3.12.2 \(\int \frac {(A+B x) (d+e x)^{9/2}}{(b x+c x^2)^3} \, dx\)
Optimal. Leaf size=461 \[ -\frac {(d+e x)^{7/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}-\frac {3 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (3 b^2 e (7 A e+4 B d)-4 b c d (9 A e+2 B d)+16 A c^2 d^2\right )}{4 b^5}+\frac {3 (c d-b e)^{5/2} \left (-b^2 c e (8 B d-A e)-4 b c^2 d (2 B d-A e)+16 A c^3 d^2-5 b^3 B e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 c^{7/2}}+\frac {(d+e x)^{3/2} \left (b c d^2 \left (-b c (13 A e+6 B d)+12 A c^2 d+2 b^2 B e\right )+x \left (b^3 c e^2 (A e+4 B d)+b^2 c^2 d e (10 A e+9 B d)-12 b c^3 d^2 (3 A e+B d)+24 A c^4 d^3-5 b^4 B e^3\right )\right )}{4 b^4 c^2 \left (b x+c x^2\right )}-\frac {3 e \sqrt {d+e x} \left (b^3 c e^2 (A e+2 B d)+b^2 c^2 d e (2 A e+3 B d)-4 b c^3 d^2 (3 A e+B d)+8 A c^4 d^3-5 b^4 B e^3\right )}{4 b^4 c^3} \]
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Rubi [A] time = 1.22, antiderivative size = 461, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used =
{818, 824, 826, 1166, 208} \begin {gather*} \frac {(d+e x)^{3/2} \left (x \left (b^2 c^2 d e (10 A e+9 B d)+b^3 c e^2 (A e+4 B d)-12 b c^3 d^2 (3 A e+B d)+24 A c^4 d^3-5 b^4 B e^3\right )+b c d^2 \left (-b c (13 A e+6 B d)+12 A c^2 d+2 b^2 B e\right )\right )}{4 b^4 c^2 \left (b x+c x^2\right )}-\frac {3 e \sqrt {d+e x} \left (b^2 c^2 d e (2 A e+3 B d)+b^3 c e^2 (A e+2 B d)-4 b c^3 d^2 (3 A e+B d)+8 A c^4 d^3-5 b^4 B e^3\right )}{4 b^4 c^3}+\frac {3 (c d-b e)^{5/2} \left (-b^2 c e (8 B d-A e)-4 b c^2 d (2 B d-A e)+16 A c^3 d^2-5 b^3 B e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 c^{7/2}}-\frac {3 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (3 b^2 e (7 A e+4 B d)-4 b c d (9 A e+2 B d)+16 A c^2 d^2\right )}{4 b^5}-\frac {(d+e x)^{7/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((A + B*x)*(d + e*x)^(9/2))/(b*x + c*x^2)^3,x]
[Out]
(-3*e*(8*A*c^4*d^3 - 5*b^4*B*e^3 + b^3*c*e^2*(2*B*d + A*e) + b^2*c^2*d*e*(3*B*d + 2*A*e) - 4*b*c^3*d^2*(B*d +
3*A*e))*Sqrt[d + e*x])/(4*b^4*c^3) - ((d + e*x)^(7/2)*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(
2*b^2*c*(b*x + c*x^2)^2) + ((d + e*x)^(3/2)*(b*c*d^2*(12*A*c^2*d + 2*b^2*B*e - b*c*(6*B*d + 13*A*e)) + (24*A*c
^4*d^3 - 5*b^4*B*e^3 + b^3*c*e^2*(4*B*d + A*e) - 12*b*c^3*d^2*(B*d + 3*A*e) + b^2*c^2*d*e*(9*B*d + 10*A*e))*x)
)/(4*b^4*c^2*(b*x + c*x^2)) - (3*d^(5/2)*(16*A*c^2*d^2 + 3*b^2*e*(4*B*d + 7*A*e) - 4*b*c*d*(2*B*d + 9*A*e))*Ar
cTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5) + (3*(c*d - b*e)^(5/2)*(16*A*c^3*d^2 - 5*b^3*B*e^2 - 4*b*c^2*d*(2*B*d -
A*e) - b^2*c*e*(8*B*d - A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*c^(7/2))
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 818
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) || !ILtQ[m + 2*p + 3, 0])
Rule 824
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(g
*(d + e*x)^m)/(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x])
/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*
e^2, 0] && FractionQ[m] && GtQ[m, 0]
Rule 826
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
Rule 1166
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^{9/2}}{\left (b x+c x^2\right )^3} \, dx &=-\frac {(d+e x)^{7/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac {\int \frac {(d+e x)^{5/2} \left (-\frac {1}{2} d \left (12 A c^2 d+2 b^2 B e-b c (6 B d+13 A e)\right )+\frac {1}{2} e \left (2 A c^2 d+5 b^2 B e-b c (B d+A e)\right ) x\right )}{\left (b x+c x^2\right )^2} \, dx}{2 b^2 c}\\ &=-\frac {(d+e x)^{7/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac {(d+e x)^{3/2} \left (b c d^2 \left (12 A c^2 d+2 b^2 B e-b c (6 B d+13 A e)\right )+\left (24 A c^4 d^3-5 b^4 B e^3+b^3 c e^2 (4 B d+A e)-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (9 B d+10 A e)\right ) x\right )}{4 b^4 c^2 \left (b x+c x^2\right )}+\frac {\int \frac {\sqrt {d+e x} \left (\frac {3}{4} c^2 d^2 \left (16 A c^2 d^2+3 b^2 e (4 B d+7 A e)-4 b c d (2 B d+9 A e)\right )-\frac {3}{4} e \left (8 A c^4 d^3-5 b^4 B e^3+b^3 c e^2 (2 B d+A e)+b^2 c^2 d e (3 B d+2 A e)-4 b c^3 d^2 (B d+3 A e)\right ) x\right )}{b x+c x^2} \, dx}{2 b^4 c^2}\\ &=-\frac {3 e \left (8 A c^4 d^3-5 b^4 B e^3+b^3 c e^2 (2 B d+A e)+b^2 c^2 d e (3 B d+2 A e)-4 b c^3 d^2 (B d+3 A e)\right ) \sqrt {d+e x}}{4 b^4 c^3}-\frac {(d+e x)^{7/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac {(d+e x)^{3/2} \left (b c d^2 \left (12 A c^2 d+2 b^2 B e-b c (6 B d+13 A e)\right )+\left (24 A c^4 d^3-5 b^4 B e^3+b^3 c e^2 (4 B d+A e)-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (9 B d+10 A e)\right ) x\right )}{4 b^4 c^2 \left (b x+c x^2\right )}+\frac {\int \frac {\frac {3}{4} c^3 d^3 \left (16 A c^2 d^2+3 b^2 e (4 B d+7 A e)-4 b c d (2 B d+9 A e)\right )+\frac {3}{4} e \left (8 A c^5 d^4-5 b^5 B e^4+b^3 c^2 d e^2 (B d+A e)+b^4 c e^3 (7 B d+A e)-4 b c^4 d^3 (B d+4 A e)+b^2 c^3 d^2 e (5 B d+7 A e)\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 c^3}\\ &=-\frac {3 e \left (8 A c^4 d^3-5 b^4 B e^3+b^3 c e^2 (2 B d+A e)+b^2 c^2 d e (3 B d+2 A e)-4 b c^3 d^2 (B d+3 A e)\right ) \sqrt {d+e x}}{4 b^4 c^3}-\frac {(d+e x)^{7/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac {(d+e x)^{3/2} \left (b c d^2 \left (12 A c^2 d+2 b^2 B e-b c (6 B d+13 A e)\right )+\left (24 A c^4 d^3-5 b^4 B e^3+b^3 c e^2 (4 B d+A e)-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (9 B d+10 A e)\right ) x\right )}{4 b^4 c^2 \left (b x+c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {3}{4} d e \left (8 A c^5 d^4-5 b^5 B e^4+b^3 c^2 d e^2 (B d+A e)+b^4 c e^3 (7 B d+A e)-4 b c^4 d^3 (B d+4 A e)+b^2 c^3 d^2 e (5 B d+7 A e)\right )+\frac {3}{4} c^3 d^3 e \left (16 A c^2 d^2+3 b^2 e (4 B d+7 A e)-4 b c d (2 B d+9 A e)\right )+\frac {3}{4} e \left (8 A c^5 d^4-5 b^5 B e^4+b^3 c^2 d e^2 (B d+A e)+b^4 c e^3 (7 B d+A e)-4 b c^4 d^3 (B d+4 A e)+b^2 c^3 d^2 e (5 B d+7 A e)\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{b^4 c^3}\\ &=-\frac {3 e \left (8 A c^4 d^3-5 b^4 B e^3+b^3 c e^2 (2 B d+A e)+b^2 c^2 d e (3 B d+2 A e)-4 b c^3 d^2 (B d+3 A e)\right ) \sqrt {d+e x}}{4 b^4 c^3}-\frac {(d+e x)^{7/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac {(d+e x)^{3/2} \left (b c d^2 \left (12 A c^2 d+2 b^2 B e-b c (6 B d+13 A e)\right )+\left (24 A c^4 d^3-5 b^4 B e^3+b^3 c e^2 (4 B d+A e)-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (9 B d+10 A e)\right ) x\right )}{4 b^4 c^2 \left (b x+c x^2\right )}-\frac {\left (3 (c d-b e)^3 \left (16 A c^3 d^2-5 b^3 B e^2-4 b c^2 d (2 B d-A e)-b^2 c e (8 B d-A e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5 c^3}+\frac {\left (3 c d^3 \left (16 A c^2 d^2+3 b^2 e (4 B d+7 A e)-4 b c d (2 B d+9 A e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5}\\ &=-\frac {3 e \left (8 A c^4 d^3-5 b^4 B e^3+b^3 c e^2 (2 B d+A e)+b^2 c^2 d e (3 B d+2 A e)-4 b c^3 d^2 (B d+3 A e)\right ) \sqrt {d+e x}}{4 b^4 c^3}-\frac {(d+e x)^{7/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac {(d+e x)^{3/2} \left (b c d^2 \left (12 A c^2 d+2 b^2 B e-b c (6 B d+13 A e)\right )+\left (24 A c^4 d^3-5 b^4 B e^3+b^3 c e^2 (4 B d+A e)-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (9 B d+10 A e)\right ) x\right )}{4 b^4 c^2 \left (b x+c x^2\right )}-\frac {3 d^{5/2} \left (16 A c^2 d^2+3 b^2 e (4 B d+7 A e)-4 b c d (2 B d+9 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {3 (c d-b e)^{5/2} \left (16 A c^3 d^2-5 b^3 B e^2-4 b c^2 d (2 B d-A e)-b^2 c e (8 B d-A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 5.22, size = 589, normalized size = 1.28 \begin {gather*} \frac {\frac {630 c (d+e x)^{11/2} \left (b^2 (-e) (7 A e+4 B d)+b c d (17 A e+6 B d)-12 A c^2 d^2\right )}{b^2 d (b e-c d)}+\frac {(b+c x) \left ((b+c x) \left (945 c^{11/2} (c d-b e)^2 \left (\frac {2}{315} \sqrt {d+e x} \left (563 d^4+506 d^3 e x+408 d^2 e^2 x^2+185 d e^3 x^3+35 e^4 x^4\right )-2 d^{9/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right ) \left (3 b^2 e (7 A e+4 B d)-4 b c d (9 A e+2 B d)+16 A c^2 d^2\right )-6 c^2 d^2 \left (b^2 c e (A e-8 B d)+4 b c^2 d (A e-2 B d)+16 A c^3 d^2-5 b^3 B e^2\right ) \left (3 (c d-b e) \left (7 (c d-b e) \left (5 (c d-b e) \left (\sqrt {c} \sqrt {d+e x} (-3 b e+4 c d+c e x)-3 (c d-b e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )\right )+3 c^{5/2} (d+e x)^{5/2}\right )+15 c^{7/2} (d+e x)^{7/2}\right )+35 c^{9/2} (d+e x)^{9/2}\right )\right )-630 b c^{13/2} (d+e x)^{11/2} \left (b^3 e^2 (7 A e+4 B d)-b^2 c d e (26 A e+9 B d)+12 b c^2 d^2 (3 A e+B d)-24 A c^3 d^3\right )\right )}{b^4 c^{11/2} d (c d-b e)^2}-\frac {630 (d+e x)^{11/2} (7 A b e-8 A c d+4 b B d)}{b d x}-\frac {1260 A (d+e x)^{11/2}}{x^2}}{2520 b d (b+c x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((A + B*x)*(d + e*x)^(9/2))/(b*x + c*x^2)^3,x]
[Out]
((630*c*(-12*A*c^2*d^2 - b^2*e*(4*B*d + 7*A*e) + b*c*d*(6*B*d + 17*A*e))*(d + e*x)^(11/2))/(b^2*d*(-(c*d) + b*
e)) - (1260*A*(d + e*x)^(11/2))/x^2 - (630*(4*b*B*d - 8*A*c*d + 7*A*b*e)*(d + e*x)^(11/2))/(b*d*x) + ((b + c*x
)*(-630*b*c^(13/2)*(-24*A*c^3*d^3 + 12*b*c^2*d^2*(B*d + 3*A*e) + b^3*e^2*(4*B*d + 7*A*e) - b^2*c*d*e*(9*B*d +
26*A*e))*(d + e*x)^(11/2) + (b + c*x)*(945*c^(11/2)*(c*d - b*e)^2*(16*A*c^2*d^2 + 3*b^2*e*(4*B*d + 7*A*e) - 4*
b*c*d*(2*B*d + 9*A*e))*((2*Sqrt[d + e*x]*(563*d^4 + 506*d^3*e*x + 408*d^2*e^2*x^2 + 185*d*e^3*x^3 + 35*e^4*x^4
))/315 - 2*d^(9/2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]) - 6*c^2*d^2*(16*A*c^3*d^2 - 5*b^3*B*e^2 + b^2*c*e*(-8*B*d +
A*e) + 4*b*c^2*d*(-2*B*d + A*e))*(35*c^(9/2)*(d + e*x)^(9/2) + 3*(c*d - b*e)*(15*c^(7/2)*(d + e*x)^(7/2) + 7*
(c*d - b*e)*(3*c^(5/2)*(d + e*x)^(5/2) + 5*(c*d - b*e)*(Sqrt[c]*Sqrt[d + e*x]*(4*c*d - 3*b*e + c*e*x) - 3*(c*d
- b*e)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])))))))/(b^4*c^(11/2)*d*(c*d - b*e)^2))/(2520*b*
d*(b + c*x)^2)
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IntegrateAlgebraic [B] time = 2.01, size = 1306, normalized size = 2.83 \begin {gather*} \frac {-24 A c^6 e \sqrt {d+e x} d^7+12 b B c^5 e \sqrt {d+e x} d^7+72 A c^6 e (d+e x)^{3/2} d^6-36 b B c^5 e (d+e x)^{3/2} d^6+84 A b c^5 e^2 \sqrt {d+e x} d^6-33 b^2 B c^4 e^2 \sqrt {d+e x} d^6-72 A c^6 e (d+e x)^{5/2} d^5+36 b B c^5 e (d+e x)^{5/2} d^5-216 A b c^5 e^2 (d+e x)^{3/2} d^5+81 b^2 B c^4 e^2 (d+e x)^{3/2} d^5-102 A b^2 c^4 e^3 \sqrt {d+e x} d^5+24 b^3 B c^3 e^3 \sqrt {d+e x} d^5+24 A c^6 e (d+e x)^{7/2} d^4-12 b B c^5 e (d+e x)^{7/2} d^4+180 A b c^5 e^2 (d+e x)^{5/2} d^4-63 b^2 B c^4 e^2 (d+e x)^{5/2} d^4+217 A b^2 c^4 e^3 (d+e x)^{3/2} d^4-41 b^3 B c^3 e^3 (d+e x)^{3/2} d^4+45 A b^3 c^3 e^4 \sqrt {d+e x} d^4+18 b^4 B c^2 e^4 \sqrt {d+e x} d^4-48 A b c^5 e^2 (d+e x)^{7/2} d^3+15 b^2 B c^4 e^2 (d+e x)^{7/2} d^3-136 A b^2 c^4 e^3 (d+e x)^{5/2} d^3+14 b^3 B c^3 e^3 (d+e x)^{5/2} d^3-74 A b^3 c^3 e^4 (d+e x)^{3/2} d^3-71 b^4 B c^2 e^4 (d+e x)^{3/2} d^3-36 b^5 B c e^5 \sqrt {d+e x} d^3+21 A b^2 c^4 e^3 (d+e x)^{7/2} d^2+3 b^3 B c^3 e^3 (d+e x)^{7/2} d^2+24 A b^3 c^3 e^4 (d+e x)^{5/2} d^2+96 b^4 B c^2 e^4 (d+e x)^{5/2} d^2-5 A b^4 c^2 e^5 (d+e x)^{3/2} d^2+97 b^5 B c e^5 (d+e x)^{3/2} d^2+15 b^6 B e^6 \sqrt {d+e x} d^2-3 A b^5 c e^6 \sqrt {d+e x} d^2+3 A b^3 c^3 e^4 (d+e x)^{7/2} d-51 b^4 B c^2 e^4 (d+e x)^{7/2} d+10 A b^4 c^2 e^5 (d+e x)^{5/2} d-86 b^5 B c e^5 (d+e x)^{5/2} d-30 b^6 B e^6 (d+e x)^{3/2} d+6 A b^5 c e^6 (d+e x)^{3/2} d+8 b^4 B c^2 e^4 (d+e x)^{9/2}-5 A b^4 c^2 e^5 (d+e x)^{7/2}+25 b^5 B c e^5 (d+e x)^{7/2}+15 b^6 B e^6 (d+e x)^{5/2}-3 A b^5 c e^6 (d+e x)^{5/2}}{4 b^4 c^3 e^2 x^2 (c d-b e-c (d+e x))^2}-\frac {3 \left (-5 B e^2 (b e-c d)^{5/2} b^3+A c e^2 (b e-c d)^{5/2} b^2-8 B c d e (b e-c d)^{5/2} b^2-8 B c^2 d^2 (b e-c d)^{5/2} b+4 A c^2 d e (b e-c d)^{5/2} b+16 A c^3 d^2 (b e-c d)^{5/2}\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {b e-c d} \sqrt {d+e x}}{c d-b e}\right )}{4 b^5 c^{7/2}}-\frac {3 \left (16 A c^2 d^{9/2}-8 b B c d^{9/2}+12 b^2 B e d^{7/2}-36 A b c e d^{7/2}+21 A b^2 e^2 d^{5/2}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[((A + B*x)*(d + e*x)^(9/2))/(b*x + c*x^2)^3,x]
[Out]
(12*b*B*c^5*d^7*e*Sqrt[d + e*x] - 24*A*c^6*d^7*e*Sqrt[d + e*x] - 33*b^2*B*c^4*d^6*e^2*Sqrt[d + e*x] + 84*A*b*c
^5*d^6*e^2*Sqrt[d + e*x] + 24*b^3*B*c^3*d^5*e^3*Sqrt[d + e*x] - 102*A*b^2*c^4*d^5*e^3*Sqrt[d + e*x] + 18*b^4*B
*c^2*d^4*e^4*Sqrt[d + e*x] + 45*A*b^3*c^3*d^4*e^4*Sqrt[d + e*x] - 36*b^5*B*c*d^3*e^5*Sqrt[d + e*x] + 15*b^6*B*
d^2*e^6*Sqrt[d + e*x] - 3*A*b^5*c*d^2*e^6*Sqrt[d + e*x] - 36*b*B*c^5*d^6*e*(d + e*x)^(3/2) + 72*A*c^6*d^6*e*(d
+ e*x)^(3/2) + 81*b^2*B*c^4*d^5*e^2*(d + e*x)^(3/2) - 216*A*b*c^5*d^5*e^2*(d + e*x)^(3/2) - 41*b^3*B*c^3*d^4*
e^3*(d + e*x)^(3/2) + 217*A*b^2*c^4*d^4*e^3*(d + e*x)^(3/2) - 71*b^4*B*c^2*d^3*e^4*(d + e*x)^(3/2) - 74*A*b^3*
c^3*d^3*e^4*(d + e*x)^(3/2) + 97*b^5*B*c*d^2*e^5*(d + e*x)^(3/2) - 5*A*b^4*c^2*d^2*e^5*(d + e*x)^(3/2) - 30*b^
6*B*d*e^6*(d + e*x)^(3/2) + 6*A*b^5*c*d*e^6*(d + e*x)^(3/2) + 36*b*B*c^5*d^5*e*(d + e*x)^(5/2) - 72*A*c^6*d^5*
e*(d + e*x)^(5/2) - 63*b^2*B*c^4*d^4*e^2*(d + e*x)^(5/2) + 180*A*b*c^5*d^4*e^2*(d + e*x)^(5/2) + 14*b^3*B*c^3*
d^3*e^3*(d + e*x)^(5/2) - 136*A*b^2*c^4*d^3*e^3*(d + e*x)^(5/2) + 96*b^4*B*c^2*d^2*e^4*(d + e*x)^(5/2) + 24*A*
b^3*c^3*d^2*e^4*(d + e*x)^(5/2) - 86*b^5*B*c*d*e^5*(d + e*x)^(5/2) + 10*A*b^4*c^2*d*e^5*(d + e*x)^(5/2) + 15*b
^6*B*e^6*(d + e*x)^(5/2) - 3*A*b^5*c*e^6*(d + e*x)^(5/2) - 12*b*B*c^5*d^4*e*(d + e*x)^(7/2) + 24*A*c^6*d^4*e*(
d + e*x)^(7/2) + 15*b^2*B*c^4*d^3*e^2*(d + e*x)^(7/2) - 48*A*b*c^5*d^3*e^2*(d + e*x)^(7/2) + 3*b^3*B*c^3*d^2*e
^3*(d + e*x)^(7/2) + 21*A*b^2*c^4*d^2*e^3*(d + e*x)^(7/2) - 51*b^4*B*c^2*d*e^4*(d + e*x)^(7/2) + 3*A*b^3*c^3*d
*e^4*(d + e*x)^(7/2) + 25*b^5*B*c*e^5*(d + e*x)^(7/2) - 5*A*b^4*c^2*e^5*(d + e*x)^(7/2) + 8*b^4*B*c^2*e^4*(d +
e*x)^(9/2))/(4*b^4*c^3*e^2*x^2*(c*d - b*e - c*(d + e*x))^2) - (3*(-8*b*B*c^2*d^2*(-(c*d) + b*e)^(5/2) + 16*A*
c^3*d^2*(-(c*d) + b*e)^(5/2) - 8*b^2*B*c*d*e*(-(c*d) + b*e)^(5/2) + 4*A*b*c^2*d*e*(-(c*d) + b*e)^(5/2) - 5*b^3
*B*e^2*(-(c*d) + b*e)^(5/2) + A*b^2*c*e^2*(-(c*d) + b*e)^(5/2))*ArcTan[(Sqrt[c]*Sqrt[-(c*d) + b*e]*Sqrt[d + e*
x])/(c*d - b*e)])/(4*b^5*c^(7/2)) - (3*(-8*b*B*c*d^(9/2) + 16*A*c^2*d^(9/2) + 12*b^2*B*d^(7/2)*e - 36*A*b*c*d^
(7/2)*e + 21*A*b^2*d^(5/2)*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5)
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fricas [B] time = 141.14, size = 3989, normalized size = 8.65
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(9/2)/(c*x^2+b*x)^3,x, algorithm="fricas")
[Out]
[1/8*(3*((8*(B*b*c^6 - 2*A*c^7)*d^4 - 4*(2*B*b^2*c^5 - 7*A*b*c^6)*d^3*e - 3*(B*b^3*c^4 + 3*A*b^2*c^5)*d^2*e^2
- 2*(B*b^4*c^3 + A*b^3*c^4)*d*e^3 + (5*B*b^5*c^2 - A*b^4*c^3)*e^4)*x^4 + 2*(8*(B*b^2*c^5 - 2*A*b*c^6)*d^4 - 4*
(2*B*b^3*c^4 - 7*A*b^2*c^5)*d^3*e - 3*(B*b^4*c^3 + 3*A*b^3*c^4)*d^2*e^2 - 2*(B*b^5*c^2 + A*b^4*c^3)*d*e^3 + (5
*B*b^6*c - A*b^5*c^2)*e^4)*x^3 + (8*(B*b^3*c^4 - 2*A*b^2*c^5)*d^4 - 4*(2*B*b^4*c^3 - 7*A*b^3*c^4)*d^3*e - 3*(B
*b^5*c^2 + 3*A*b^4*c^3)*d^2*e^2 - 2*(B*b^6*c + A*b^5*c^2)*d*e^3 + (5*B*b^7 - A*b^6*c)*e^4)*x^2)*sqrt((c*d - b*
e)/c)*log((c*e*x + 2*c*d - b*e - 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) + 3*((21*A*b^2*c^5*d^2*e^2
- 8*(B*b*c^6 - 2*A*c^7)*d^4 + 12*(B*b^2*c^5 - 3*A*b*c^6)*d^3*e)*x^4 + 2*(21*A*b^3*c^4*d^2*e^2 - 8*(B*b^2*c^5 -
2*A*b*c^6)*d^4 + 12*(B*b^3*c^4 - 3*A*b^2*c^5)*d^3*e)*x^3 + (21*A*b^4*c^3*d^2*e^2 - 8*(B*b^3*c^4 - 2*A*b^2*c^5
)*d^4 + 12*(B*b^4*c^3 - 3*A*b^3*c^4)*d^3*e)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(8*B
*b^5*c^2*e^4*x^4 - 2*A*b^4*c^3*d^4 - (12*(B*b^2*c^5 - 2*A*b*c^6)*d^4 - 3*(5*B*b^3*c^4 - 16*A*b^2*c^5)*d^3*e -
3*(B*b^4*c^3 + 7*A*b^3*c^4)*d^2*e^2 + (19*B*b^5*c^2 - 3*A*b^4*c^3)*d*e^3 - 5*(5*B*b^6*c - A*b^5*c^2)*e^4)*x^3
- (18*(B*b^3*c^4 - 2*A*b^2*c^5)*d^4 - (23*B*b^4*c^3 - 73*A*b^3*c^4)*d^3*e + 3*(3*B*b^5*c^2 - 11*A*b^4*c^3)*d^2
*e^2 + (11*B*b^6*c + 5*A*b^5*c^2)*d*e^3 - 3*(5*B*b^7 - A*b^6*c)*e^4)*x^2 - (17*A*b^4*c^3*d^3*e + 4*(B*b^4*c^3
- 2*A*b^3*c^4)*d^4)*x)*sqrt(e*x + d))/(b^5*c^5*x^4 + 2*b^6*c^4*x^3 + b^7*c^3*x^2), -1/8*(6*((8*(B*b*c^6 - 2*A*
c^7)*d^4 - 4*(2*B*b^2*c^5 - 7*A*b*c^6)*d^3*e - 3*(B*b^3*c^4 + 3*A*b^2*c^5)*d^2*e^2 - 2*(B*b^4*c^3 + A*b^3*c^4)
*d*e^3 + (5*B*b^5*c^2 - A*b^4*c^3)*e^4)*x^4 + 2*(8*(B*b^2*c^5 - 2*A*b*c^6)*d^4 - 4*(2*B*b^3*c^4 - 7*A*b^2*c^5)
*d^3*e - 3*(B*b^4*c^3 + 3*A*b^3*c^4)*d^2*e^2 - 2*(B*b^5*c^2 + A*b^4*c^3)*d*e^3 + (5*B*b^6*c - A*b^5*c^2)*e^4)*
x^3 + (8*(B*b^3*c^4 - 2*A*b^2*c^5)*d^4 - 4*(2*B*b^4*c^3 - 7*A*b^3*c^4)*d^3*e - 3*(B*b^5*c^2 + 3*A*b^4*c^3)*d^2
*e^2 - 2*(B*b^6*c + A*b^5*c^2)*d*e^3 + (5*B*b^7 - A*b^6*c)*e^4)*x^2)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d
)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) - 3*((21*A*b^2*c^5*d^2*e^2 - 8*(B*b*c^6 - 2*A*c^7)*d^4 + 12*(B*b^2*c^5 -
3*A*b*c^6)*d^3*e)*x^4 + 2*(21*A*b^3*c^4*d^2*e^2 - 8*(B*b^2*c^5 - 2*A*b*c^6)*d^4 + 12*(B*b^3*c^4 - 3*A*b^2*c^5
)*d^3*e)*x^3 + (21*A*b^4*c^3*d^2*e^2 - 8*(B*b^3*c^4 - 2*A*b^2*c^5)*d^4 + 12*(B*b^4*c^3 - 3*A*b^3*c^4)*d^3*e)*x
^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(8*B*b^5*c^2*e^4*x^4 - 2*A*b^4*c^3*d^4 - (12*(B*b
^2*c^5 - 2*A*b*c^6)*d^4 - 3*(5*B*b^3*c^4 - 16*A*b^2*c^5)*d^3*e - 3*(B*b^4*c^3 + 7*A*b^3*c^4)*d^2*e^2 + (19*B*b
^5*c^2 - 3*A*b^4*c^3)*d*e^3 - 5*(5*B*b^6*c - A*b^5*c^2)*e^4)*x^3 - (18*(B*b^3*c^4 - 2*A*b^2*c^5)*d^4 - (23*B*b
^4*c^3 - 73*A*b^3*c^4)*d^3*e + 3*(3*B*b^5*c^2 - 11*A*b^4*c^3)*d^2*e^2 + (11*B*b^6*c + 5*A*b^5*c^2)*d*e^3 - 3*(
5*B*b^7 - A*b^6*c)*e^4)*x^2 - (17*A*b^4*c^3*d^3*e + 4*(B*b^4*c^3 - 2*A*b^3*c^4)*d^4)*x)*sqrt(e*x + d))/(b^5*c^
5*x^4 + 2*b^6*c^4*x^3 + b^7*c^3*x^2), 1/8*(6*((21*A*b^2*c^5*d^2*e^2 - 8*(B*b*c^6 - 2*A*c^7)*d^4 + 12*(B*b^2*c^
5 - 3*A*b*c^6)*d^3*e)*x^4 + 2*(21*A*b^3*c^4*d^2*e^2 - 8*(B*b^2*c^5 - 2*A*b*c^6)*d^4 + 12*(B*b^3*c^4 - 3*A*b^2*
c^5)*d^3*e)*x^3 + (21*A*b^4*c^3*d^2*e^2 - 8*(B*b^3*c^4 - 2*A*b^2*c^5)*d^4 + 12*(B*b^4*c^3 - 3*A*b^3*c^4)*d^3*e
)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + 3*((8*(B*b*c^6 - 2*A*c^7)*d^4 - 4*(2*B*b^2*c^5 - 7*A*b*c^6)
*d^3*e - 3*(B*b^3*c^4 + 3*A*b^2*c^5)*d^2*e^2 - 2*(B*b^4*c^3 + A*b^3*c^4)*d*e^3 + (5*B*b^5*c^2 - A*b^4*c^3)*e^4
)*x^4 + 2*(8*(B*b^2*c^5 - 2*A*b*c^6)*d^4 - 4*(2*B*b^3*c^4 - 7*A*b^2*c^5)*d^3*e - 3*(B*b^4*c^3 + 3*A*b^3*c^4)*d
^2*e^2 - 2*(B*b^5*c^2 + A*b^4*c^3)*d*e^3 + (5*B*b^6*c - A*b^5*c^2)*e^4)*x^3 + (8*(B*b^3*c^4 - 2*A*b^2*c^5)*d^4
- 4*(2*B*b^4*c^3 - 7*A*b^3*c^4)*d^3*e - 3*(B*b^5*c^2 + 3*A*b^4*c^3)*d^2*e^2 - 2*(B*b^6*c + A*b^5*c^2)*d*e^3 +
(5*B*b^7 - A*b^6*c)*e^4)*x^2)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e - 2*sqrt(e*x + d)*c*sqrt((c*d - b*
e)/c))/(c*x + b)) + 2*(8*B*b^5*c^2*e^4*x^4 - 2*A*b^4*c^3*d^4 - (12*(B*b^2*c^5 - 2*A*b*c^6)*d^4 - 3*(5*B*b^3*c^
4 - 16*A*b^2*c^5)*d^3*e - 3*(B*b^4*c^3 + 7*A*b^3*c^4)*d^2*e^2 + (19*B*b^5*c^2 - 3*A*b^4*c^3)*d*e^3 - 5*(5*B*b^
6*c - A*b^5*c^2)*e^4)*x^3 - (18*(B*b^3*c^4 - 2*A*b^2*c^5)*d^4 - (23*B*b^4*c^3 - 73*A*b^3*c^4)*d^3*e + 3*(3*B*b
^5*c^2 - 11*A*b^4*c^3)*d^2*e^2 + (11*B*b^6*c + 5*A*b^5*c^2)*d*e^3 - 3*(5*B*b^7 - A*b^6*c)*e^4)*x^2 - (17*A*b^4
*c^3*d^3*e + 4*(B*b^4*c^3 - 2*A*b^3*c^4)*d^4)*x)*sqrt(e*x + d))/(b^5*c^5*x^4 + 2*b^6*c^4*x^3 + b^7*c^3*x^2), -
1/4*(3*((8*(B*b*c^6 - 2*A*c^7)*d^4 - 4*(2*B*b^2*c^5 - 7*A*b*c^6)*d^3*e - 3*(B*b^3*c^4 + 3*A*b^2*c^5)*d^2*e^2 -
2*(B*b^4*c^3 + A*b^3*c^4)*d*e^3 + (5*B*b^5*c^2 - A*b^4*c^3)*e^4)*x^4 + 2*(8*(B*b^2*c^5 - 2*A*b*c^6)*d^4 - 4*(
2*B*b^3*c^4 - 7*A*b^2*c^5)*d^3*e - 3*(B*b^4*c^3 + 3*A*b^3*c^4)*d^2*e^2 - 2*(B*b^5*c^2 + A*b^4*c^3)*d*e^3 + (5*
B*b^6*c - A*b^5*c^2)*e^4)*x^3 + (8*(B*b^3*c^4 - 2*A*b^2*c^5)*d^4 - 4*(2*B*b^4*c^3 - 7*A*b^3*c^4)*d^3*e - 3*(B*
b^5*c^2 + 3*A*b^4*c^3)*d^2*e^2 - 2*(B*b^6*c + A*b^5*c^2)*d*e^3 + (5*B*b^7 - A*b^6*c)*e^4)*x^2)*sqrt(-(c*d - b*
e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) - 3*((21*A*b^2*c^5*d^2*e^2 - 8*(B*b*c^6 - 2*A*
c^7)*d^4 + 12*(B*b^2*c^5 - 3*A*b*c^6)*d^3*e)*x^4 + 2*(21*A*b^3*c^4*d^2*e^2 - 8*(B*b^2*c^5 - 2*A*b*c^6)*d^4 + 1
2*(B*b^3*c^4 - 3*A*b^2*c^5)*d^3*e)*x^3 + (21*A*b^4*c^3*d^2*e^2 - 8*(B*b^3*c^4 - 2*A*b^2*c^5)*d^4 + 12*(B*b^4*c
^3 - 3*A*b^3*c^4)*d^3*e)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - (8*B*b^5*c^2*e^4*x^4 - 2*A*b^4*c^3*d
^4 - (12*(B*b^2*c^5 - 2*A*b*c^6)*d^4 - 3*(5*B*b^3*c^4 - 16*A*b^2*c^5)*d^3*e - 3*(B*b^4*c^3 + 7*A*b^3*c^4)*d^2*
e^2 + (19*B*b^5*c^2 - 3*A*b^4*c^3)*d*e^3 - 5*(5*B*b^6*c - A*b^5*c^2)*e^4)*x^3 - (18*(B*b^3*c^4 - 2*A*b^2*c^5)*
d^4 - (23*B*b^4*c^3 - 73*A*b^3*c^4)*d^3*e + 3*(3*B*b^5*c^2 - 11*A*b^4*c^3)*d^2*e^2 + (11*B*b^6*c + 5*A*b^5*c^2
)*d*e^3 - 3*(5*B*b^7 - A*b^6*c)*e^4)*x^2 - (17*A*b^4*c^3*d^3*e + 4*(B*b^4*c^3 - 2*A*b^3*c^4)*d^4)*x)*sqrt(e*x
+ d))/(b^5*c^5*x^4 + 2*b^6*c^4*x^3 + b^7*c^3*x^2)]
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giac [B] time = 0.36, size = 1245, normalized size = 2.70
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(9/2)/(c*x^2+b*x)^3,x, algorithm="giac")
[Out]
2*sqrt(x*e + d)*B*e^4/c^3 - 3/4*(8*B*b*c*d^5 - 16*A*c^2*d^5 - 12*B*b^2*d^4*e + 36*A*b*c*d^4*e - 21*A*b^2*d^3*e
^2)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^5*sqrt(-d)) + 3/4*(8*B*b*c^5*d^5 - 16*A*c^6*d^5 - 16*B*b^2*c^4*d^4*e + 4
4*A*b*c^5*d^4*e + 5*B*b^3*c^3*d^3*e^2 - 37*A*b^2*c^4*d^3*e^2 + B*b^4*c^2*d^2*e^3 + 7*A*b^3*c^3*d^2*e^3 + 7*B*b
^5*c*d*e^4 + A*b^4*c^2*d*e^4 - 5*B*b^6*e^5 + A*b^5*c*e^5)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-
c^2*d + b*c*e)*b^5*c^3) - 1/4*(12*(x*e + d)^(7/2)*B*b*c^5*d^4*e - 24*(x*e + d)^(7/2)*A*c^6*d^4*e - 36*(x*e + d
)^(5/2)*B*b*c^5*d^5*e + 72*(x*e + d)^(5/2)*A*c^6*d^5*e + 36*(x*e + d)^(3/2)*B*b*c^5*d^6*e - 72*(x*e + d)^(3/2)
*A*c^6*d^6*e - 12*sqrt(x*e + d)*B*b*c^5*d^7*e + 24*sqrt(x*e + d)*A*c^6*d^7*e - 15*(x*e + d)^(7/2)*B*b^2*c^4*d^
3*e^2 + 48*(x*e + d)^(7/2)*A*b*c^5*d^3*e^2 + 63*(x*e + d)^(5/2)*B*b^2*c^4*d^4*e^2 - 180*(x*e + d)^(5/2)*A*b*c^
5*d^4*e^2 - 81*(x*e + d)^(3/2)*B*b^2*c^4*d^5*e^2 + 216*(x*e + d)^(3/2)*A*b*c^5*d^5*e^2 + 33*sqrt(x*e + d)*B*b^
2*c^4*d^6*e^2 - 84*sqrt(x*e + d)*A*b*c^5*d^6*e^2 - 3*(x*e + d)^(7/2)*B*b^3*c^3*d^2*e^3 - 21*(x*e + d)^(7/2)*A*
b^2*c^4*d^2*e^3 - 14*(x*e + d)^(5/2)*B*b^3*c^3*d^3*e^3 + 136*(x*e + d)^(5/2)*A*b^2*c^4*d^3*e^3 + 41*(x*e + d)^
(3/2)*B*b^3*c^3*d^4*e^3 - 217*(x*e + d)^(3/2)*A*b^2*c^4*d^4*e^3 - 24*sqrt(x*e + d)*B*b^3*c^3*d^5*e^3 + 102*sqr
t(x*e + d)*A*b^2*c^4*d^5*e^3 + 19*(x*e + d)^(7/2)*B*b^4*c^2*d*e^4 - 3*(x*e + d)^(7/2)*A*b^3*c^3*d*e^4 - 48*(x*
e + d)^(5/2)*B*b^4*c^2*d^2*e^4 - 24*(x*e + d)^(5/2)*A*b^3*c^3*d^2*e^4 + 39*(x*e + d)^(3/2)*B*b^4*c^2*d^3*e^4 +
74*(x*e + d)^(3/2)*A*b^3*c^3*d^3*e^4 - 10*sqrt(x*e + d)*B*b^4*c^2*d^4*e^4 - 45*sqrt(x*e + d)*A*b^3*c^3*d^4*e^
4 - 9*(x*e + d)^(7/2)*B*b^5*c*e^5 + 5*(x*e + d)^(7/2)*A*b^4*c^2*e^5 + 38*(x*e + d)^(5/2)*B*b^5*c*d*e^5 - 10*(x
*e + d)^(5/2)*A*b^4*c^2*d*e^5 - 49*(x*e + d)^(3/2)*B*b^5*c*d^2*e^5 + 5*(x*e + d)^(3/2)*A*b^4*c^2*d^2*e^5 + 20*
sqrt(x*e + d)*B*b^5*c*d^3*e^5 - 7*(x*e + d)^(5/2)*B*b^6*e^6 + 3*(x*e + d)^(5/2)*A*b^5*c*e^6 + 14*(x*e + d)^(3/
2)*B*b^6*d*e^6 - 6*(x*e + d)^(3/2)*A*b^5*c*d*e^6 - 7*sqrt(x*e + d)*B*b^6*d^2*e^6 + 3*sqrt(x*e + d)*A*b^5*c*d^2
*e^6)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)^2*b^4*c^3)
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maple [B] time = 0.09, size = 1421, normalized size = 3.08
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)*(e*x+d)^(9/2)/(c*x^2+b*x)^3,x)
[Out]
2*e^4*B/c^3*(e*x+d)^(1/2)+5/2*e^4/c/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)*d^2+15/2*e^4/b/(c*e*x+b*e)^2*A*(e*x+d)^(1/2)
*d^2+3/4*e^4/b/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A*d-19/4*e^4/c/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B*d+3/4*e^3/b/(c*e*x+b*e
)^2*(e*x+d)^(3/2)*B*d^2+21/4*e^3/b^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d^2+5*e
^3/b/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)*d^3+6/b^4*c^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*
c)*B*d^5-12/b^5*c^3/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d^5+9/4*e^5*b/c^2/(c*e*x
+b*e)^2*(e*x+d)^(3/2)*B-3/4*e^6*b/c^2/(c*e*x+b*e)^2*A*(e*x+d)^(1/2)-15/4*e^5*b/c^3/((b*e-c*d)*c)^(1/2)*arctan(
(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B+7/4*e^6*b^2/c^3/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)-1/e*d^4/b^3/x^2*(e*x+d)^(
3/2)*B+1/e*d^5/b^3/x^2*(e*x+d)^(1/2)*B+27*e*d^(7/2)/b^4*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c+21/4*e^4/c^2/((b*e-
c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*d+15/4*e^2/b^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^
(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*d^3+3/4*e^5/c^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)
*A-17/4*d^3/b^3/x^2*(e*x+d)^(3/2)*A+15/4*d^4/b^3/x^2*(e*x+d)^(1/2)*A-12*d^(9/2)/b^5*arctanh((e*x+d)^(1/2)/d^(1
/2))*A*c^2+6*d^(9/2)/b^4*arctanh((e*x+d)^(1/2)/d^(1/2))*B*c-5/4*e^5/c/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A-63/4*e^2*d
^(5/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))*A-9*e*d^(7/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))*B-111/4*e^2/b^3*c/(
(b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d^3+33*e/b^4*c^2/((b*e-c*d)*c)^(1/2)*arctan((
e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d^4+3/4*e^4/b/c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(
1/2)*c)*A*d-25/4*e^2/b^2*c/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)*d^4+2*e/b^3*c^2/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)*d^5+45/
4*e^2/b^3*c^2/(c*e*x+b*e)^2*A*(e*x+d)^(1/2)*d^4-3*e/b^4*c^3/(c*e*x+b*e)^2*A*(e*x+d)^(1/2)*d^5-5*e^5*b/c^2/(c*e
*x+b*e)^2*B*(e*x+d)^(1/2)*d-15*e^3/b^2*c/(c*e*x+b*e)^2*A*(e*x+d)^(1/2)*d^3+21/4*e^3/b^2*c/(c*e*x+b*e)^2*(e*x+d
)^(3/2)*A*d^2-31/4*e^2/b^3*c^2/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A*d^3-12*e/b^3*c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)
^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*d^4+3*e/b^4*c^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A*d^4+15/4*e^2/b^2*c/(c*e*x+b*e)^2
*(e*x+d)^(3/2)*B*d^3-2*e/b^3*c^2/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B*d^4+3/e*d^4/b^4/x^2*(e*x+d)^(3/2)*A*c-3/e*d^5/b
^4/x^2*(e*x+d)^(1/2)*A*c+3/4*e^3/b/c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*d^2
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(9/2)/(c*x^2+b*x)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d positive or negative?
________________________________________________________________________________________
mupad [B] time = 5.05, size = 16542, normalized size = 35.88
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((A + B*x)*(d + e*x)^(9/2))/(b*x + c*x^2)^3,x)
[Out]
atan(((((3*(64*A*b^14*c^5*d*e^7 - 320*B*b^15*c^4*d*e^7 - 512*A*b^10*c^9*d^5*e^3 + 1280*A*b^11*c^8*d^4*e^4 - 89
6*A*b^12*c^7*d^3*e^5 + 64*A*b^13*c^6*d^2*e^6 + 256*B*b^11*c^8*d^5*e^3 - 448*B*b^12*c^7*d^4*e^4 + 64*B*b^13*c^6
*d^3*e^5 + 448*B*b^14*c^5*d^2*e^6))/(64*b^12*c^5) - ((64*b^11*c^7*e^3 - 128*b^10*c^8*d*e^2)*(d + e*x)^(1/2)*((
9*(256*A^2*c^11*d^9 - 25*B^2*b^11*e^9 - A^2*b^9*c^2*e^9 + 64*B^2*b^2*c^9*d^9 + 1968*A^2*b^2*c^9*d^7*e^2 - 1512
*A^2*b^3*c^8*d^6*e^3 + 441*A^2*b^4*c^7*d^5*e^4 - 21*A^2*b^5*c^6*d^4*e^5 + 42*A^2*b^6*c^5*d^3*e^6 - 18*A^2*b^7*
c^4*d^2*e^7 + 144*B^2*b^4*c^7*d^7*e^2 + 105*B^2*b^6*c^5*d^5*e^4 - 189*B^2*b^7*c^4*d^4*e^5 + 42*B^2*b^8*c^3*d^3
*e^6 + 6*B^2*b^9*c^2*d^2*e^7 - 1152*A^2*b*c^10*d^8*e + 45*B^2*b^10*c*d*e^8 - 3*A^2*b^8*c^3*d*e^8 - 192*B^2*b^3
*c^8*d^8*e - 256*A*B*b*c^10*d^9 + 10*A*B*b^10*c*e^9 + 960*A*B*b^2*c^9*d^8*e + 6*A*B*b^9*c^2*d*e^8 - 1200*A*B*b
^3*c^8*d^7*e^2 + 504*A*B*b^4*c^7*d^6*e^3 - 210*A*B*b^5*c^6*d^5*e^4 + 546*A*B*b^6*c^5*d^4*e^5 - 420*A*B*b^7*c^4
*d^3*e^6 + 60*A*B*b^8*c^3*d^2*e^7))/(64*b^10*c^7))^(1/2))/(8*b^8*c^5))*((9*(256*A^2*c^11*d^9 - 25*B^2*b^11*e^9
- A^2*b^9*c^2*e^9 + 64*B^2*b^2*c^9*d^9 + 1968*A^2*b^2*c^9*d^7*e^2 - 1512*A^2*b^3*c^8*d^6*e^3 + 441*A^2*b^4*c^
7*d^5*e^4 - 21*A^2*b^5*c^6*d^4*e^5 + 42*A^2*b^6*c^5*d^3*e^6 - 18*A^2*b^7*c^4*d^2*e^7 + 144*B^2*b^4*c^7*d^7*e^2
+ 105*B^2*b^6*c^5*d^5*e^4 - 189*B^2*b^7*c^4*d^4*e^5 + 42*B^2*b^8*c^3*d^3*e^6 + 6*B^2*b^9*c^2*d^2*e^7 - 1152*A
^2*b*c^10*d^8*e + 45*B^2*b^10*c*d*e^8 - 3*A^2*b^8*c^3*d*e^8 - 192*B^2*b^3*c^8*d^8*e - 256*A*B*b*c^10*d^9 + 10*
A*B*b^10*c*e^9 + 960*A*B*b^2*c^9*d^8*e + 6*A*B*b^9*c^2*d*e^8 - 1200*A*B*b^3*c^8*d^7*e^2 + 504*A*B*b^4*c^7*d^6*
e^3 - 210*A*B*b^5*c^6*d^5*e^4 + 546*A*B*b^6*c^5*d^4*e^5 - 420*A*B*b^7*c^4*d^3*e^6 + 60*A*B*b^8*c^3*d^2*e^7))/(
64*b^10*c^7))^(1/2) - ((d + e*x)^(1/2)*(225*B^2*b^12*e^12 + 9*A^2*b^10*c^2*e^12 + 4608*A^2*c^12*d^10*e^2 + 457
92*A^2*b^2*c^10*d^8*e^4 - 44928*A^2*b^3*c^9*d^7*e^5 + 21546*A^2*b^4*c^8*d^6*e^6 - 4158*A^2*b^5*c^7*d^5*e^7 + 5
67*A^2*b^6*c^6*d^4*e^8 - 540*A^2*b^7*c^5*d^3*e^9 + 135*A^2*b^8*c^4*d^2*e^10 + 1152*B^2*b^2*c^10*d^10*e^2 - 403
2*B^2*b^3*c^9*d^9*e^3 + 4320*B^2*b^4*c^8*d^8*e^4 - 1296*B^2*b^5*c^7*d^7*e^5 + 945*B^2*b^6*c^6*d^6*e^6 - 2646*B
^2*b^7*c^5*d^5*e^7 + 2079*B^2*b^8*c^4*d^4*e^8 - 324*B^2*b^9*c^3*d^3*e^9 + 351*B^2*b^10*c^2*d^2*e^10 - 630*B^2*
b^11*c*d*e^11 - 23040*A^2*b*c^11*d^9*e^3 + 18*A^2*b^9*c^3*d*e^11 - 90*A*B*b^11*c*e^12 - 4608*A*B*b*c^11*d^10*e
^2 + 36*A*B*b^10*c^2*d*e^11 + 19584*A*B*b^2*c^10*d^9*e^3 - 30240*A*B*b^3*c^9*d^8*e^4 + 19872*A*B*b^4*c^8*d^7*e
^5 - 6426*A*B*b^5*c^7*d^6*e^6 + 6804*A*B*b^6*c^6*d^5*e^7 - 8694*A*B*b^7*c^5*d^4*e^8 + 4320*A*B*b^8*c^4*d^3*e^9
- 486*A*B*b^9*c^3*d^2*e^10))/(8*b^8*c^5))*((9*(256*A^2*c^11*d^9 - 25*B^2*b^11*e^9 - A^2*b^9*c^2*e^9 + 64*B^2*
b^2*c^9*d^9 + 1968*A^2*b^2*c^9*d^7*e^2 - 1512*A^2*b^3*c^8*d^6*e^3 + 441*A^2*b^4*c^7*d^5*e^4 - 21*A^2*b^5*c^6*d
^4*e^5 + 42*A^2*b^6*c^5*d^3*e^6 - 18*A^2*b^7*c^4*d^2*e^7 + 144*B^2*b^4*c^7*d^7*e^2 + 105*B^2*b^6*c^5*d^5*e^4 -
189*B^2*b^7*c^4*d^4*e^5 + 42*B^2*b^8*c^3*d^3*e^6 + 6*B^2*b^9*c^2*d^2*e^7 - 1152*A^2*b*c^10*d^8*e + 45*B^2*b^1
0*c*d*e^8 - 3*A^2*b^8*c^3*d*e^8 - 192*B^2*b^3*c^8*d^8*e - 256*A*B*b*c^10*d^9 + 10*A*B*b^10*c*e^9 + 960*A*B*b^2
*c^9*d^8*e + 6*A*B*b^9*c^2*d*e^8 - 1200*A*B*b^3*c^8*d^7*e^2 + 504*A*B*b^4*c^7*d^6*e^3 - 210*A*B*b^5*c^6*d^5*e^
4 + 546*A*B*b^6*c^5*d^4*e^5 - 420*A*B*b^7*c^4*d^3*e^6 + 60*A*B*b^8*c^3*d^2*e^7))/(64*b^10*c^7))^(1/2)*1i - (((
3*(64*A*b^14*c^5*d*e^7 - 320*B*b^15*c^4*d*e^7 - 512*A*b^10*c^9*d^5*e^3 + 1280*A*b^11*c^8*d^4*e^4 - 896*A*b^12*
c^7*d^3*e^5 + 64*A*b^13*c^6*d^2*e^6 + 256*B*b^11*c^8*d^5*e^3 - 448*B*b^12*c^7*d^4*e^4 + 64*B*b^13*c^6*d^3*e^5
+ 448*B*b^14*c^5*d^2*e^6))/(64*b^12*c^5) + ((64*b^11*c^7*e^3 - 128*b^10*c^8*d*e^2)*(d + e*x)^(1/2)*((9*(256*A^
2*c^11*d^9 - 25*B^2*b^11*e^9 - A^2*b^9*c^2*e^9 + 64*B^2*b^2*c^9*d^9 + 1968*A^2*b^2*c^9*d^7*e^2 - 1512*A^2*b^3*
c^8*d^6*e^3 + 441*A^2*b^4*c^7*d^5*e^4 - 21*A^2*b^5*c^6*d^4*e^5 + 42*A^2*b^6*c^5*d^3*e^6 - 18*A^2*b^7*c^4*d^2*e
^7 + 144*B^2*b^4*c^7*d^7*e^2 + 105*B^2*b^6*c^5*d^5*e^4 - 189*B^2*b^7*c^4*d^4*e^5 + 42*B^2*b^8*c^3*d^3*e^6 + 6*
B^2*b^9*c^2*d^2*e^7 - 1152*A^2*b*c^10*d^8*e + 45*B^2*b^10*c*d*e^8 - 3*A^2*b^8*c^3*d*e^8 - 192*B^2*b^3*c^8*d^8*
e - 256*A*B*b*c^10*d^9 + 10*A*B*b^10*c*e^9 + 960*A*B*b^2*c^9*d^8*e + 6*A*B*b^9*c^2*d*e^8 - 1200*A*B*b^3*c^8*d^
7*e^2 + 504*A*B*b^4*c^7*d^6*e^3 - 210*A*B*b^5*c^6*d^5*e^4 + 546*A*B*b^6*c^5*d^4*e^5 - 420*A*B*b^7*c^4*d^3*e^6
+ 60*A*B*b^8*c^3*d^2*e^7))/(64*b^10*c^7))^(1/2))/(8*b^8*c^5))*((9*(256*A^2*c^11*d^9 - 25*B^2*b^11*e^9 - A^2*b^
9*c^2*e^9 + 64*B^2*b^2*c^9*d^9 + 1968*A^2*b^2*c^9*d^7*e^2 - 1512*A^2*b^3*c^8*d^6*e^3 + 441*A^2*b^4*c^7*d^5*e^4
- 21*A^2*b^5*c^6*d^4*e^5 + 42*A^2*b^6*c^5*d^3*e^6 - 18*A^2*b^7*c^4*d^2*e^7 + 144*B^2*b^4*c^7*d^7*e^2 + 105*B^
2*b^6*c^5*d^5*e^4 - 189*B^2*b^7*c^4*d^4*e^5 + 42*B^2*b^8*c^3*d^3*e^6 + 6*B^2*b^9*c^2*d^2*e^7 - 1152*A^2*b*c^10
*d^8*e + 45*B^2*b^10*c*d*e^8 - 3*A^2*b^8*c^3*d*e^8 - 192*B^2*b^3*c^8*d^8*e - 256*A*B*b*c^10*d^9 + 10*A*B*b^10*
c*e^9 + 960*A*B*b^2*c^9*d^8*e + 6*A*B*b^9*c^2*d*e^8 - 1200*A*B*b^3*c^8*d^7*e^2 + 504*A*B*b^4*c^7*d^6*e^3 - 210
*A*B*b^5*c^6*d^5*e^4 + 546*A*B*b^6*c^5*d^4*e^5 - 420*A*B*b^7*c^4*d^3*e^6 + 60*A*B*b^8*c^3*d^2*e^7))/(64*b^10*c
^7))^(1/2) + ((d + e*x)^(1/2)*(225*B^2*b^12*e^12 + 9*A^2*b^10*c^2*e^12 + 4608*A^2*c^12*d^10*e^2 + 45792*A^2*b^
2*c^10*d^8*e^4 - 44928*A^2*b^3*c^9*d^7*e^5 + 21546*A^2*b^4*c^8*d^6*e^6 - 4158*A^2*b^5*c^7*d^5*e^7 + 567*A^2*b^
6*c^6*d^4*e^8 - 540*A^2*b^7*c^5*d^3*e^9 + 135*A^2*b^8*c^4*d^2*e^10 + 1152*B^2*b^2*c^10*d^10*e^2 - 4032*B^2*b^3
*c^9*d^9*e^3 + 4320*B^2*b^4*c^8*d^8*e^4 - 1296*B^2*b^5*c^7*d^7*e^5 + 945*B^2*b^6*c^6*d^6*e^6 - 2646*B^2*b^7*c^
5*d^5*e^7 + 2079*B^2*b^8*c^4*d^4*e^8 - 324*B^2*b^9*c^3*d^3*e^9 + 351*B^2*b^10*c^2*d^2*e^10 - 630*B^2*b^11*c*d*
e^11 - 23040*A^2*b*c^11*d^9*e^3 + 18*A^2*b^9*c^3*d*e^11 - 90*A*B*b^11*c*e^12 - 4608*A*B*b*c^11*d^10*e^2 + 36*A
*B*b^10*c^2*d*e^11 + 19584*A*B*b^2*c^10*d^9*e^3 - 30240*A*B*b^3*c^9*d^8*e^4 + 19872*A*B*b^4*c^8*d^7*e^5 - 6426
*A*B*b^5*c^7*d^6*e^6 + 6804*A*B*b^6*c^6*d^5*e^7 - 8694*A*B*b^7*c^5*d^4*e^8 + 4320*A*B*b^8*c^4*d^3*e^9 - 486*A*
B*b^9*c^3*d^2*e^10))/(8*b^8*c^5))*((9*(256*A^2*c^11*d^9 - 25*B^2*b^11*e^9 - A^2*b^9*c^2*e^9 + 64*B^2*b^2*c^9*d
^9 + 1968*A^2*b^2*c^9*d^7*e^2 - 1512*A^2*b^3*c^8*d^6*e^3 + 441*A^2*b^4*c^7*d^5*e^4 - 21*A^2*b^5*c^6*d^4*e^5 +
42*A^2*b^6*c^5*d^3*e^6 - 18*A^2*b^7*c^4*d^2*e^7 + 144*B^2*b^4*c^7*d^7*e^2 + 105*B^2*b^6*c^5*d^5*e^4 - 189*B^2*
b^7*c^4*d^4*e^5 + 42*B^2*b^8*c^3*d^3*e^6 + 6*B^2*b^9*c^2*d^2*e^7 - 1152*A^2*b*c^10*d^8*e + 45*B^2*b^10*c*d*e^8
- 3*A^2*b^8*c^3*d*e^8 - 192*B^2*b^3*c^8*d^8*e - 256*A*B*b*c^10*d^9 + 10*A*B*b^10*c*e^9 + 960*A*B*b^2*c^9*d^8*
e + 6*A*B*b^9*c^2*d*e^8 - 1200*A*B*b^3*c^8*d^7*e^2 + 504*A*B*b^4*c^7*d^6*e^3 - 210*A*B*b^5*c^6*d^5*e^4 + 546*A
*B*b^6*c^5*d^4*e^5 - 420*A*B*b^7*c^4*d^3*e^6 + 60*A*B*b^8*c^3*d^2*e^7))/(64*b^10*c^7))^(1/2)*1i)/((((3*(64*A*b
^14*c^5*d*e^7 - 320*B*b^15*c^4*d*e^7 - 512*A*b^10*c^9*d^5*e^3 + 1280*A*b^11*c^8*d^4*e^4 - 896*A*b^12*c^7*d^3*e
^5 + 64*A*b^13*c^6*d^2*e^6 + 256*B*b^11*c^8*d^5*e^3 - 448*B*b^12*c^7*d^4*e^4 + 64*B*b^13*c^6*d^3*e^5 + 448*B*b
^14*c^5*d^2*e^6))/(64*b^12*c^5) - ((64*b^11*c^7*e^3 - 128*b^10*c^8*d*e^2)*(d + e*x)^(1/2)*((9*(256*A^2*c^11*d^
9 - 25*B^2*b^11*e^9 - A^2*b^9*c^2*e^9 + 64*B^2*b^2*c^9*d^9 + 1968*A^2*b^2*c^9*d^7*e^2 - 1512*A^2*b^3*c^8*d^6*e
^3 + 441*A^2*b^4*c^7*d^5*e^4 - 21*A^2*b^5*c^6*d^4*e^5 + 42*A^2*b^6*c^5*d^3*e^6 - 18*A^2*b^7*c^4*d^2*e^7 + 144*
B^2*b^4*c^7*d^7*e^2 + 105*B^2*b^6*c^5*d^5*e^4 - 189*B^2*b^7*c^4*d^4*e^5 + 42*B^2*b^8*c^3*d^3*e^6 + 6*B^2*b^9*c
^2*d^2*e^7 - 1152*A^2*b*c^10*d^8*e + 45*B^2*b^10*c*d*e^8 - 3*A^2*b^8*c^3*d*e^8 - 192*B^2*b^3*c^8*d^8*e - 256*A
*B*b*c^10*d^9 + 10*A*B*b^10*c*e^9 + 960*A*B*b^2*c^9*d^8*e + 6*A*B*b^9*c^2*d*e^8 - 1200*A*B*b^3*c^8*d^7*e^2 + 5
04*A*B*b^4*c^7*d^6*e^3 - 210*A*B*b^5*c^6*d^5*e^4 + 546*A*B*b^6*c^5*d^4*e^5 - 420*A*B*b^7*c^4*d^3*e^6 + 60*A*B*
b^8*c^3*d^2*e^7))/(64*b^10*c^7))^(1/2))/(8*b^8*c^5))*((9*(256*A^2*c^11*d^9 - 25*B^2*b^11*e^9 - A^2*b^9*c^2*e^9
+ 64*B^2*b^2*c^9*d^9 + 1968*A^2*b^2*c^9*d^7*e^2 - 1512*A^2*b^3*c^8*d^6*e^3 + 441*A^2*b^4*c^7*d^5*e^4 - 21*A^2
*b^5*c^6*d^4*e^5 + 42*A^2*b^6*c^5*d^3*e^6 - 18*A^2*b^7*c^4*d^2*e^7 + 144*B^2*b^4*c^7*d^7*e^2 + 105*B^2*b^6*c^5
*d^5*e^4 - 189*B^2*b^7*c^4*d^4*e^5 + 42*B^2*b^8*c^3*d^3*e^6 + 6*B^2*b^9*c^2*d^2*e^7 - 1152*A^2*b*c^10*d^8*e +
45*B^2*b^10*c*d*e^8 - 3*A^2*b^8*c^3*d*e^8 - 192*B^2*b^3*c^8*d^8*e - 256*A*B*b*c^10*d^9 + 10*A*B*b^10*c*e^9 + 9
60*A*B*b^2*c^9*d^8*e + 6*A*B*b^9*c^2*d*e^8 - 1200*A*B*b^3*c^8*d^7*e^2 + 504*A*B*b^4*c^7*d^6*e^3 - 210*A*B*b^5*
c^6*d^5*e^4 + 546*A*B*b^6*c^5*d^4*e^5 - 420*A*B*b^7*c^4*d^3*e^6 + 60*A*B*b^8*c^3*d^2*e^7))/(64*b^10*c^7))^(1/2
) - ((d + e*x)^(1/2)*(225*B^2*b^12*e^12 + 9*A^2*b^10*c^2*e^12 + 4608*A^2*c^12*d^10*e^2 + 45792*A^2*b^2*c^10*d^
8*e^4 - 44928*A^2*b^3*c^9*d^7*e^5 + 21546*A^2*b^4*c^8*d^6*e^6 - 4158*A^2*b^5*c^7*d^5*e^7 + 567*A^2*b^6*c^6*d^4
*e^8 - 540*A^2*b^7*c^5*d^3*e^9 + 135*A^2*b^8*c^4*d^2*e^10 + 1152*B^2*b^2*c^10*d^10*e^2 - 4032*B^2*b^3*c^9*d^9*
e^3 + 4320*B^2*b^4*c^8*d^8*e^4 - 1296*B^2*b^5*c^7*d^7*e^5 + 945*B^2*b^6*c^6*d^6*e^6 - 2646*B^2*b^7*c^5*d^5*e^7
+ 2079*B^2*b^8*c^4*d^4*e^8 - 324*B^2*b^9*c^3*d^3*e^9 + 351*B^2*b^10*c^2*d^2*e^10 - 630*B^2*b^11*c*d*e^11 - 23
040*A^2*b*c^11*d^9*e^3 + 18*A^2*b^9*c^3*d*e^11 - 90*A*B*b^11*c*e^12 - 4608*A*B*b*c^11*d^10*e^2 + 36*A*B*b^10*c
^2*d*e^11 + 19584*A*B*b^2*c^10*d^9*e^3 - 30240*A*B*b^3*c^9*d^8*e^4 + 19872*A*B*b^4*c^8*d^7*e^5 - 6426*A*B*b^5*
c^7*d^6*e^6 + 6804*A*B*b^6*c^6*d^5*e^7 - 8694*A*B*b^7*c^5*d^4*e^8 + 4320*A*B*b^8*c^4*d^3*e^9 - 486*A*B*b^9*c^3
*d^2*e^10))/(8*b^8*c^5))*((9*(256*A^2*c^11*d^9 - 25*B^2*b^11*e^9 - A^2*b^9*c^2*e^9 + 64*B^2*b^2*c^9*d^9 + 1968
*A^2*b^2*c^9*d^7*e^2 - 1512*A^2*b^3*c^8*d^6*e^3 + 441*A^2*b^4*c^7*d^5*e^4 - 21*A^2*b^5*c^6*d^4*e^5 + 42*A^2*b^
6*c^5*d^3*e^6 - 18*A^2*b^7*c^4*d^2*e^7 + 144*B^2*b^4*c^7*d^7*e^2 + 105*B^2*b^6*c^5*d^5*e^4 - 189*B^2*b^7*c^4*d
^4*e^5 + 42*B^2*b^8*c^3*d^3*e^6 + 6*B^2*b^9*c^2*d^2*e^7 - 1152*A^2*b*c^10*d^8*e + 45*B^2*b^10*c*d*e^8 - 3*A^2*
b^8*c^3*d*e^8 - 192*B^2*b^3*c^8*d^8*e - 256*A*B*b*c^10*d^9 + 10*A*B*b^10*c*e^9 + 960*A*B*b^2*c^9*d^8*e + 6*A*B
*b^9*c^2*d*e^8 - 1200*A*B*b^3*c^8*d^7*e^2 + 504*A*B*b^4*c^7*d^6*e^3 - 210*A*B*b^5*c^6*d^5*e^4 + 546*A*B*b^6*c^
5*d^4*e^5 - 420*A*B*b^7*c^4*d^3*e^6 + 60*A*B*b^8*c^3*d^2*e^7))/(64*b^10*c^7))^(1/2) + (((3*(64*A*b^14*c^5*d*e^
7 - 320*B*b^15*c^4*d*e^7 - 512*A*b^10*c^9*d^5*e^3 + 1280*A*b^11*c^8*d^4*e^4 - 896*A*b^12*c^7*d^3*e^5 + 64*A*b^
13*c^6*d^2*e^6 + 256*B*b^11*c^8*d^5*e^3 - 448*B*b^12*c^7*d^4*e^4 + 64*B*b^13*c^6*d^3*e^5 + 448*B*b^14*c^5*d^2*
e^6))/(64*b^12*c^5) + ((64*b^11*c^7*e^3 - 128*b^10*c^8*d*e^2)*(d + e*x)^(1/2)*((9*(256*A^2*c^11*d^9 - 25*B^2*b
^11*e^9 - A^2*b^9*c^2*e^9 + 64*B^2*b^2*c^9*d^9 + 1968*A^2*b^2*c^9*d^7*e^2 - 1512*A^2*b^3*c^8*d^6*e^3 + 441*A^2
*b^4*c^7*d^5*e^4 - 21*A^2*b^5*c^6*d^4*e^5 + 42*A^2*b^6*c^5*d^3*e^6 - 18*A^2*b^7*c^4*d^2*e^7 + 144*B^2*b^4*c^7*
d^7*e^2 + 105*B^2*b^6*c^5*d^5*e^4 - 189*B^2*b^7*c^4*d^4*e^5 + 42*B^2*b^8*c^3*d^3*e^6 + 6*B^2*b^9*c^2*d^2*e^7 -
1152*A^2*b*c^10*d^8*e + 45*B^2*b^10*c*d*e^8 - 3*A^2*b^8*c^3*d*e^8 - 192*B^2*b^3*c^8*d^8*e - 256*A*B*b*c^10*d^
9 + 10*A*B*b^10*c*e^9 + 960*A*B*b^2*c^9*d^8*e + 6*A*B*b^9*c^2*d*e^8 - 1200*A*B*b^3*c^8*d^7*e^2 + 504*A*B*b^4*c
^7*d^6*e^3 - 210*A*B*b^5*c^6*d^5*e^4 + 546*A*B*b^6*c^5*d^4*e^5 - 420*A*B*b^7*c^4*d^3*e^6 + 60*A*B*b^8*c^3*d^2*
e^7))/(64*b^10*c^7))^(1/2))/(8*b^8*c^5))*((9*(256*A^2*c^11*d^9 - 25*B^2*b^11*e^9 - A^2*b^9*c^2*e^9 + 64*B^2*b^
2*c^9*d^9 + 1968*A^2*b^2*c^9*d^7*e^2 - 1512*A^2*b^3*c^8*d^6*e^3 + 441*A^2*b^4*c^7*d^5*e^4 - 21*A^2*b^5*c^6*d^4
*e^5 + 42*A^2*b^6*c^5*d^3*e^6 - 18*A^2*b^7*c^4*d^2*e^7 + 144*B^2*b^4*c^7*d^7*e^2 + 105*B^2*b^6*c^5*d^5*e^4 - 1
89*B^2*b^7*c^4*d^4*e^5 + 42*B^2*b^8*c^3*d^3*e^6 + 6*B^2*b^9*c^2*d^2*e^7 - 1152*A^2*b*c^10*d^8*e + 45*B^2*b^10*
c*d*e^8 - 3*A^2*b^8*c^3*d*e^8 - 192*B^2*b^3*c^8*d^8*e - 256*A*B*b*c^10*d^9 + 10*A*B*b^10*c*e^9 + 960*A*B*b^2*c
^9*d^8*e + 6*A*B*b^9*c^2*d*e^8 - 1200*A*B*b^3*c^8*d^7*e^2 + 504*A*B*b^4*c^7*d^6*e^3 - 210*A*B*b^5*c^6*d^5*e^4
+ 546*A*B*b^6*c^5*d^4*e^5 - 420*A*B*b^7*c^4*d^3*e^6 + 60*A*B*b^8*c^3*d^2*e^7))/(64*b^10*c^7))^(1/2) + ((d + e*
x)^(1/2)*(225*B^2*b^12*e^12 + 9*A^2*b^10*c^2*e^12 + 4608*A^2*c^12*d^10*e^2 + 45792*A^2*b^2*c^10*d^8*e^4 - 4492
8*A^2*b^3*c^9*d^7*e^5 + 21546*A^2*b^4*c^8*d^6*e^6 - 4158*A^2*b^5*c^7*d^5*e^7 + 567*A^2*b^6*c^6*d^4*e^8 - 540*A
^2*b^7*c^5*d^3*e^9 + 135*A^2*b^8*c^4*d^2*e^10 + 1152*B^2*b^2*c^10*d^10*e^2 - 4032*B^2*b^3*c^9*d^9*e^3 + 4320*B
^2*b^4*c^8*d^8*e^4 - 1296*B^2*b^5*c^7*d^7*e^5 + 945*B^2*b^6*c^6*d^6*e^6 - 2646*B^2*b^7*c^5*d^5*e^7 + 2079*B^2*
b^8*c^4*d^4*e^8 - 324*B^2*b^9*c^3*d^3*e^9 + 351*B^2*b^10*c^2*d^2*e^10 - 630*B^2*b^11*c*d*e^11 - 23040*A^2*b*c^
11*d^9*e^3 + 18*A^2*b^9*c^3*d*e^11 - 90*A*B*b^11*c*e^12 - 4608*A*B*b*c^11*d^10*e^2 + 36*A*B*b^10*c^2*d*e^11 +
19584*A*B*b^2*c^10*d^9*e^3 - 30240*A*B*b^3*c^9*d^8*e^4 + 19872*A*B*b^4*c^8*d^7*e^5 - 6426*A*B*b^5*c^7*d^6*e^6
+ 6804*A*B*b^6*c^6*d^5*e^7 - 8694*A*B*b^7*c^5*d^4*e^8 + 4320*A*B*b^8*c^4*d^3*e^9 - 486*A*B*b^9*c^3*d^2*e^10))/
(8*b^8*c^5))*((9*(256*A^2*c^11*d^9 - 25*B^2*b^11*e^9 - A^2*b^9*c^2*e^9 + 64*B^2*b^2*c^9*d^9 + 1968*A^2*b^2*c^9
*d^7*e^2 - 1512*A^2*b^3*c^8*d^6*e^3 + 441*A^2*b^4*c^7*d^5*e^4 - 21*A^2*b^5*c^6*d^4*e^5 + 42*A^2*b^6*c^5*d^3*e^
6 - 18*A^2*b^7*c^4*d^2*e^7 + 144*B^2*b^4*c^7*d^7*e^2 + 105*B^2*b^6*c^5*d^5*e^4 - 189*B^2*b^7*c^4*d^4*e^5 + 42*
B^2*b^8*c^3*d^3*e^6 + 6*B^2*b^9*c^2*d^2*e^7 - 1152*A^2*b*c^10*d^8*e + 45*B^2*b^10*c*d*e^8 - 3*A^2*b^8*c^3*d*e^
8 - 192*B^2*b^3*c^8*d^8*e - 256*A*B*b*c^10*d^9 + 10*A*B*b^10*c*e^9 + 960*A*B*b^2*c^9*d^8*e + 6*A*B*b^9*c^2*d*e
^8 - 1200*A*B*b^3*c^8*d^7*e^2 + 504*A*B*b^4*c^7*d^6*e^3 - 210*A*B*b^5*c^6*d^5*e^4 + 546*A*B*b^6*c^5*d^4*e^5 -
420*A*B*b^7*c^4*d^3*e^6 + 60*A*B*b^8*c^3*d^2*e^7))/(64*b^10*c^7))^(1/2) - (3*(2700*B^3*b^13*d^4*e^13 - 18432*A
^3*c^13*d^14*e^3 - 381312*A^3*b^2*c^11*d^12*e^5 + 610560*A^3*b^3*c^10*d^11*e^6 - 562968*A^3*b^4*c^9*d^10*e^7 +
293112*A^3*b^5*c^8*d^9*e^8 - 84483*A^3*b^6*c^7*d^8*e^9 + 23868*A^3*b^7*c^6*d^7*e^10 - 11943*A^3*b^8*c^5*d^6*e
^11 + 2331*A^3*b^9*c^4*d^5*e^12 + 54*A^3*b^10*c^3*d^4*e^13 + 189*A^3*b^11*c^2*d^3*e^14 + 2304*B^3*b^3*c^10*d^1
4*e^3 - 10944*B^3*b^4*c^9*d^13*e^4 + 17856*B^3*b^5*c^8*d^12*e^5 - 14328*B^3*b^6*c^7*d^11*e^6 + 18828*B^3*b^7*c
^6*d^10*e^7 - 30672*B^3*b^8*c^5*d^9*e^8 + 21060*B^3*b^9*c^4*d^8*e^9 - 6696*B^3*b^10*c^3*d^7*e^10 + 9252*B^3*b^
11*c^2*d^6*e^11 + 4725*A*B^2*b^13*d^3*e^14 + 129024*A^3*b*c^12*d^13*e^4 - 9360*B^3*b^12*c*d^5*e^12 - 13824*A*B
^2*b^2*c^11*d^14*e^3 + 76032*A*B^2*b^3*c^10*d^13*e^4 - 158976*A*B^2*b^4*c^9*d^12*e^5 + 167184*A*B^2*b^5*c^8*d^
11*e^6 - 143316*A*B^2*b^6*c^7*d^10*e^7 + 184869*A*B^2*b^7*c^6*d^9*e^8 - 187434*A*B^2*b^8*c^5*d^8*e^9 + 93987*A
*B^2*b^9*c^4*d^7*e^10 - 35640*A*B^2*b^10*c^3*d^6*e^11 + 34803*A*B^2*b^11*c^2*d^5*e^12 - 172800*A^2*B*b^2*c^11*
d^13*e^4 + 437184*A^2*B*b^3*c^10*d^12*e^5 - 578448*A^2*B*b^4*c^9*d^11*e^6 + 471420*A^2*B*b^5*c^8*d^10*e^7 - 36
1773*A^2*B*b^6*c^7*d^9*e^8 + 350001*A^2*B*b^7*c^6*d^8*e^9 - 253071*A^2*B*b^8*c^5*d^7*e^10 + 90423*A^2*B*b^9*c^
4*d^6*e^11 - 12798*A^2*B*b^10*c^3*d^5*e^12 + 4104*A^2*B*b^11*c^2*d^4*e^13 - 22410*A*B^2*b^12*c*d^4*e^13 + 2764
8*A^2*B*b*c^12*d^14*e^3 - 1890*A^2*B*b^12*c*d^3*e^14))/(32*b^12*c^5)))*((9*(256*A^2*c^11*d^9 - 25*B^2*b^11*e^9
- A^2*b^9*c^2*e^9 + 64*B^2*b^2*c^9*d^9 + 1968*A^2*b^2*c^9*d^7*e^2 - 1512*A^2*b^3*c^8*d^6*e^3 + 441*A^2*b^4*c^
7*d^5*e^4 - 21*A^2*b^5*c^6*d^4*e^5 + 42*A^2*b^6*c^5*d^3*e^6 - 18*A^2*b^7*c^4*d^2*e^7 + 144*B^2*b^4*c^7*d^7*e^2
+ 105*B^2*b^6*c^5*d^5*e^4 - 189*B^2*b^7*c^4*d^4*e^5 + 42*B^2*b^8*c^3*d^3*e^6 + 6*B^2*b^9*c^2*d^2*e^7 - 1152*A
^2*b*c^10*d^8*e + 45*B^2*b^10*c*d*e^8 - 3*A^2*b^8*c^3*d*e^8 - 192*B^2*b^3*c^8*d^8*e - 256*A*B*b*c^10*d^9 + 10*
A*B*b^10*c*e^9 + 960*A*B*b^2*c^9*d^8*e + 6*A*B*b^9*c^2*d*e^8 - 1200*A*B*b^3*c^8*d^7*e^2 + 504*A*B*b^4*c^7*d^6*
e^3 - 210*A*B*b^5*c^6*d^5*e^4 + 546*A*B*b^6*c^5*d^4*e^5 - 420*A*B*b^7*c^4*d^3*e^6 + 60*A*B*b^8*c^3*d^2*e^7))/(
64*b^10*c^7))^(1/2)*2i + atan(((((3*(64*A*b^14*c^5*d*e^7 - 320*B*b^15*c^4*d*e^7 - 512*A*b^10*c^9*d^5*e^3 + 128
0*A*b^11*c^8*d^4*e^4 - 896*A*b^12*c^7*d^3*e^5 + 64*A*b^13*c^6*d^2*e^6 + 256*B*b^11*c^8*d^5*e^3 - 448*B*b^12*c^
7*d^4*e^4 + 64*B*b^13*c^6*d^3*e^5 + 448*B*b^14*c^5*d^2*e^6))/(64*b^12*c^5) - ((64*b^11*c^7*e^3 - 128*b^10*c^8*
d*e^2)*(d + e*x)^(1/2)*((9*(256*A^2*c^4*d^9 + 64*B^2*b^2*c^2*d^9 + 441*A^2*b^4*d^5*e^4 + 144*B^2*b^4*d^7*e^2 +
1968*A^2*b^2*c^2*d^7*e^2 + 504*A*B*b^4*d^6*e^3 - 1152*A^2*b*c^3*d^8*e - 192*B^2*b^3*c*d^8*e - 1512*A^2*b^3*c*
d^6*e^3 - 256*A*B*b*c^3*d^9 + 960*A*B*b^2*c^2*d^8*e - 1200*A*B*b^3*c*d^7*e^2))/(64*b^10))^(1/2))/(8*b^8*c^5))*
((9*(256*A^2*c^4*d^9 + 64*B^2*b^2*c^2*d^9 + 441*A^2*b^4*d^5*e^4 + 144*B^2*b^4*d^7*e^2 + 1968*A^2*b^2*c^2*d^7*e
^2 + 504*A*B*b^4*d^6*e^3 - 1152*A^2*b*c^3*d^8*e - 192*B^2*b^3*c*d^8*e - 1512*A^2*b^3*c*d^6*e^3 - 256*A*B*b*c^3
*d^9 + 960*A*B*b^2*c^2*d^8*e - 1200*A*B*b^3*c*d^7*e^2))/(64*b^10))^(1/2) - ((d + e*x)^(1/2)*(225*B^2*b^12*e^12
+ 9*A^2*b^10*c^2*e^12 + 4608*A^2*c^12*d^10*e^2 + 45792*A^2*b^2*c^10*d^8*e^4 - 44928*A^2*b^3*c^9*d^7*e^5 + 215
46*A^2*b^4*c^8*d^6*e^6 - 4158*A^2*b^5*c^7*d^5*e^7 + 567*A^2*b^6*c^6*d^4*e^8 - 540*A^2*b^7*c^5*d^3*e^9 + 135*A^
2*b^8*c^4*d^2*e^10 + 1152*B^2*b^2*c^10*d^10*e^2 - 4032*B^2*b^3*c^9*d^9*e^3 + 4320*B^2*b^4*c^8*d^8*e^4 - 1296*B
^2*b^5*c^7*d^7*e^5 + 945*B^2*b^6*c^6*d^6*e^6 - 2646*B^2*b^7*c^5*d^5*e^7 + 2079*B^2*b^8*c^4*d^4*e^8 - 324*B^2*b
^9*c^3*d^3*e^9 + 351*B^2*b^10*c^2*d^2*e^10 - 630*B^2*b^11*c*d*e^11 - 23040*A^2*b*c^11*d^9*e^3 + 18*A^2*b^9*c^3
*d*e^11 - 90*A*B*b^11*c*e^12 - 4608*A*B*b*c^11*d^10*e^2 + 36*A*B*b^10*c^2*d*e^11 + 19584*A*B*b^2*c^10*d^9*e^3
- 30240*A*B*b^3*c^9*d^8*e^4 + 19872*A*B*b^4*c^8*d^7*e^5 - 6426*A*B*b^5*c^7*d^6*e^6 + 6804*A*B*b^6*c^6*d^5*e^7
- 8694*A*B*b^7*c^5*d^4*e^8 + 4320*A*B*b^8*c^4*d^3*e^9 - 486*A*B*b^9*c^3*d^2*e^10))/(8*b^8*c^5))*((9*(256*A^2*c
^4*d^9 + 64*B^2*b^2*c^2*d^9 + 441*A^2*b^4*d^5*e^4 + 144*B^2*b^4*d^7*e^2 + 1968*A^2*b^2*c^2*d^7*e^2 + 504*A*B*b
^4*d^6*e^3 - 1152*A^2*b*c^3*d^8*e - 192*B^2*b^3*c*d^8*e - 1512*A^2*b^3*c*d^6*e^3 - 256*A*B*b*c^3*d^9 + 960*A*B
*b^2*c^2*d^8*e - 1200*A*B*b^3*c*d^7*e^2))/(64*b^10))^(1/2)*1i - (((3*(64*A*b^14*c^5*d*e^7 - 320*B*b^15*c^4*d*e
^7 - 512*A*b^10*c^9*d^5*e^3 + 1280*A*b^11*c^8*d^4*e^4 - 896*A*b^12*c^7*d^3*e^5 + 64*A*b^13*c^6*d^2*e^6 + 256*B
*b^11*c^8*d^5*e^3 - 448*B*b^12*c^7*d^4*e^4 + 64*B*b^13*c^6*d^3*e^5 + 448*B*b^14*c^5*d^2*e^6))/(64*b^12*c^5) +
((64*b^11*c^7*e^3 - 128*b^10*c^8*d*e^2)*(d + e*x)^(1/2)*((9*(256*A^2*c^4*d^9 + 64*B^2*b^2*c^2*d^9 + 441*A^2*b^
4*d^5*e^4 + 144*B^2*b^4*d^7*e^2 + 1968*A^2*b^2*c^2*d^7*e^2 + 504*A*B*b^4*d^6*e^3 - 1152*A^2*b*c^3*d^8*e - 192*
B^2*b^3*c*d^8*e - 1512*A^2*b^3*c*d^6*e^3 - 256*A*B*b*c^3*d^9 + 960*A*B*b^2*c^2*d^8*e - 1200*A*B*b^3*c*d^7*e^2)
)/(64*b^10))^(1/2))/(8*b^8*c^5))*((9*(256*A^2*c^4*d^9 + 64*B^2*b^2*c^2*d^9 + 441*A^2*b^4*d^5*e^4 + 144*B^2*b^4
*d^7*e^2 + 1968*A^2*b^2*c^2*d^7*e^2 + 504*A*B*b^4*d^6*e^3 - 1152*A^2*b*c^3*d^8*e - 192*B^2*b^3*c*d^8*e - 1512*
A^2*b^3*c*d^6*e^3 - 256*A*B*b*c^3*d^9 + 960*A*B*b^2*c^2*d^8*e - 1200*A*B*b^3*c*d^7*e^2))/(64*b^10))^(1/2) + ((
d + e*x)^(1/2)*(225*B^2*b^12*e^12 + 9*A^2*b^10*c^2*e^12 + 4608*A^2*c^12*d^10*e^2 + 45792*A^2*b^2*c^10*d^8*e^4
- 44928*A^2*b^3*c^9*d^7*e^5 + 21546*A^2*b^4*c^8*d^6*e^6 - 4158*A^2*b^5*c^7*d^5*e^7 + 567*A^2*b^6*c^6*d^4*e^8 -
540*A^2*b^7*c^5*d^3*e^9 + 135*A^2*b^8*c^4*d^2*e^10 + 1152*B^2*b^2*c^10*d^10*e^2 - 4032*B^2*b^3*c^9*d^9*e^3 +
4320*B^2*b^4*c^8*d^8*e^4 - 1296*B^2*b^5*c^7*d^7*e^5 + 945*B^2*b^6*c^6*d^6*e^6 - 2646*B^2*b^7*c^5*d^5*e^7 + 207
9*B^2*b^8*c^4*d^4*e^8 - 324*B^2*b^9*c^3*d^3*e^9 + 351*B^2*b^10*c^2*d^2*e^10 - 630*B^2*b^11*c*d*e^11 - 23040*A^
2*b*c^11*d^9*e^3 + 18*A^2*b^9*c^3*d*e^11 - 90*A*B*b^11*c*e^12 - 4608*A*B*b*c^11*d^10*e^2 + 36*A*B*b^10*c^2*d*e
^11 + 19584*A*B*b^2*c^10*d^9*e^3 - 30240*A*B*b^3*c^9*d^8*e^4 + 19872*A*B*b^4*c^8*d^7*e^5 - 6426*A*B*b^5*c^7*d^
6*e^6 + 6804*A*B*b^6*c^6*d^5*e^7 - 8694*A*B*b^7*c^5*d^4*e^8 + 4320*A*B*b^8*c^4*d^3*e^9 - 486*A*B*b^9*c^3*d^2*e
^10))/(8*b^8*c^5))*((9*(256*A^2*c^4*d^9 + 64*B^2*b^2*c^2*d^9 + 441*A^2*b^4*d^5*e^4 + 144*B^2*b^4*d^7*e^2 + 196
8*A^2*b^2*c^2*d^7*e^2 + 504*A*B*b^4*d^6*e^3 - 1152*A^2*b*c^3*d^8*e - 192*B^2*b^3*c*d^8*e - 1512*A^2*b^3*c*d^6*
e^3 - 256*A*B*b*c^3*d^9 + 960*A*B*b^2*c^2*d^8*e - 1200*A*B*b^3*c*d^7*e^2))/(64*b^10))^(1/2)*1i)/((((3*(64*A*b^
14*c^5*d*e^7 - 320*B*b^15*c^4*d*e^7 - 512*A*b^10*c^9*d^5*e^3 + 1280*A*b^11*c^8*d^4*e^4 - 896*A*b^12*c^7*d^3*e^
5 + 64*A*b^13*c^6*d^2*e^6 + 256*B*b^11*c^8*d^5*e^3 - 448*B*b^12*c^7*d^4*e^4 + 64*B*b^13*c^6*d^3*e^5 + 448*B*b^
14*c^5*d^2*e^6))/(64*b^12*c^5) - ((64*b^11*c^7*e^3 - 128*b^10*c^8*d*e^2)*(d + e*x)^(1/2)*((9*(256*A^2*c^4*d^9
+ 64*B^2*b^2*c^2*d^9 + 441*A^2*b^4*d^5*e^4 + 144*B^2*b^4*d^7*e^2 + 1968*A^2*b^2*c^2*d^7*e^2 + 504*A*B*b^4*d^6*
e^3 - 1152*A^2*b*c^3*d^8*e - 192*B^2*b^3*c*d^8*e - 1512*A^2*b^3*c*d^6*e^3 - 256*A*B*b*c^3*d^9 + 960*A*B*b^2*c^
2*d^8*e - 1200*A*B*b^3*c*d^7*e^2))/(64*b^10))^(1/2))/(8*b^8*c^5))*((9*(256*A^2*c^4*d^9 + 64*B^2*b^2*c^2*d^9 +
441*A^2*b^4*d^5*e^4 + 144*B^2*b^4*d^7*e^2 + 1968*A^2*b^2*c^2*d^7*e^2 + 504*A*B*b^4*d^6*e^3 - 1152*A^2*b*c^3*d^
8*e - 192*B^2*b^3*c*d^8*e - 1512*A^2*b^3*c*d^6*e^3 - 256*A*B*b*c^3*d^9 + 960*A*B*b^2*c^2*d^8*e - 1200*A*B*b^3*
c*d^7*e^2))/(64*b^10))^(1/2) - ((d + e*x)^(1/2)*(225*B^2*b^12*e^12 + 9*A^2*b^10*c^2*e^12 + 4608*A^2*c^12*d^10*
e^2 + 45792*A^2*b^2*c^10*d^8*e^4 - 44928*A^2*b^3*c^9*d^7*e^5 + 21546*A^2*b^4*c^8*d^6*e^6 - 4158*A^2*b^5*c^7*d^
5*e^7 + 567*A^2*b^6*c^6*d^4*e^8 - 540*A^2*b^7*c^5*d^3*e^9 + 135*A^2*b^8*c^4*d^2*e^10 + 1152*B^2*b^2*c^10*d^10*
e^2 - 4032*B^2*b^3*c^9*d^9*e^3 + 4320*B^2*b^4*c^8*d^8*e^4 - 1296*B^2*b^5*c^7*d^7*e^5 + 945*B^2*b^6*c^6*d^6*e^6
- 2646*B^2*b^7*c^5*d^5*e^7 + 2079*B^2*b^8*c^4*d^4*e^8 - 324*B^2*b^9*c^3*d^3*e^9 + 351*B^2*b^10*c^2*d^2*e^10 -
630*B^2*b^11*c*d*e^11 - 23040*A^2*b*c^11*d^9*e^3 + 18*A^2*b^9*c^3*d*e^11 - 90*A*B*b^11*c*e^12 - 4608*A*B*b*c^
11*d^10*e^2 + 36*A*B*b^10*c^2*d*e^11 + 19584*A*B*b^2*c^10*d^9*e^3 - 30240*A*B*b^3*c^9*d^8*e^4 + 19872*A*B*b^4*
c^8*d^7*e^5 - 6426*A*B*b^5*c^7*d^6*e^6 + 6804*A*B*b^6*c^6*d^5*e^7 - 8694*A*B*b^7*c^5*d^4*e^8 + 4320*A*B*b^8*c^
4*d^3*e^9 - 486*A*B*b^9*c^3*d^2*e^10))/(8*b^8*c^5))*((9*(256*A^2*c^4*d^9 + 64*B^2*b^2*c^2*d^9 + 441*A^2*b^4*d^
5*e^4 + 144*B^2*b^4*d^7*e^2 + 1968*A^2*b^2*c^2*d^7*e^2 + 504*A*B*b^4*d^6*e^3 - 1152*A^2*b*c^3*d^8*e - 192*B^2*
b^3*c*d^8*e - 1512*A^2*b^3*c*d^6*e^3 - 256*A*B*b*c^3*d^9 + 960*A*B*b^2*c^2*d^8*e - 1200*A*B*b^3*c*d^7*e^2))/(6
4*b^10))^(1/2) + (((3*(64*A*b^14*c^5*d*e^7 - 320*B*b^15*c^4*d*e^7 - 512*A*b^10*c^9*d^5*e^3 + 1280*A*b^11*c^8*d
^4*e^4 - 896*A*b^12*c^7*d^3*e^5 + 64*A*b^13*c^6*d^2*e^6 + 256*B*b^11*c^8*d^5*e^3 - 448*B*b^12*c^7*d^4*e^4 + 64
*B*b^13*c^6*d^3*e^5 + 448*B*b^14*c^5*d^2*e^6))/(64*b^12*c^5) + ((64*b^11*c^7*e^3 - 128*b^10*c^8*d*e^2)*(d + e*
x)^(1/2)*((9*(256*A^2*c^4*d^9 + 64*B^2*b^2*c^2*d^9 + 441*A^2*b^4*d^5*e^4 + 144*B^2*b^4*d^7*e^2 + 1968*A^2*b^2*
c^2*d^7*e^2 + 504*A*B*b^4*d^6*e^3 - 1152*A^2*b*c^3*d^8*e - 192*B^2*b^3*c*d^8*e - 1512*A^2*b^3*c*d^6*e^3 - 256*
A*B*b*c^3*d^9 + 960*A*B*b^2*c^2*d^8*e - 1200*A*B*b^3*c*d^7*e^2))/(64*b^10))^(1/2))/(8*b^8*c^5))*((9*(256*A^2*c
^4*d^9 + 64*B^2*b^2*c^2*d^9 + 441*A^2*b^4*d^5*e^4 + 144*B^2*b^4*d^7*e^2 + 1968*A^2*b^2*c^2*d^7*e^2 + 504*A*B*b
^4*d^6*e^3 - 1152*A^2*b*c^3*d^8*e - 192*B^2*b^3*c*d^8*e - 1512*A^2*b^3*c*d^6*e^3 - 256*A*B*b*c^3*d^9 + 960*A*B
*b^2*c^2*d^8*e - 1200*A*B*b^3*c*d^7*e^2))/(64*b^10))^(1/2) + ((d + e*x)^(1/2)*(225*B^2*b^12*e^12 + 9*A^2*b^10*
c^2*e^12 + 4608*A^2*c^12*d^10*e^2 + 45792*A^2*b^2*c^10*d^8*e^4 - 44928*A^2*b^3*c^9*d^7*e^5 + 21546*A^2*b^4*c^8
*d^6*e^6 - 4158*A^2*b^5*c^7*d^5*e^7 + 567*A^2*b^6*c^6*d^4*e^8 - 540*A^2*b^7*c^5*d^3*e^9 + 135*A^2*b^8*c^4*d^2*
e^10 + 1152*B^2*b^2*c^10*d^10*e^2 - 4032*B^2*b^3*c^9*d^9*e^3 + 4320*B^2*b^4*c^8*d^8*e^4 - 1296*B^2*b^5*c^7*d^7
*e^5 + 945*B^2*b^6*c^6*d^6*e^6 - 2646*B^2*b^7*c^5*d^5*e^7 + 2079*B^2*b^8*c^4*d^4*e^8 - 324*B^2*b^9*c^3*d^3*e^9
+ 351*B^2*b^10*c^2*d^2*e^10 - 630*B^2*b^11*c*d*e^11 - 23040*A^2*b*c^11*d^9*e^3 + 18*A^2*b^9*c^3*d*e^11 - 90*A
*B*b^11*c*e^12 - 4608*A*B*b*c^11*d^10*e^2 + 36*A*B*b^10*c^2*d*e^11 + 19584*A*B*b^2*c^10*d^9*e^3 - 30240*A*B*b^
3*c^9*d^8*e^4 + 19872*A*B*b^4*c^8*d^7*e^5 - 6426*A*B*b^5*c^7*d^6*e^6 + 6804*A*B*b^6*c^6*d^5*e^7 - 8694*A*B*b^7
*c^5*d^4*e^8 + 4320*A*B*b^8*c^4*d^3*e^9 - 486*A*B*b^9*c^3*d^2*e^10))/(8*b^8*c^5))*((9*(256*A^2*c^4*d^9 + 64*B^
2*b^2*c^2*d^9 + 441*A^2*b^4*d^5*e^4 + 144*B^2*b^4*d^7*e^2 + 1968*A^2*b^2*c^2*d^7*e^2 + 504*A*B*b^4*d^6*e^3 - 1
152*A^2*b*c^3*d^8*e - 192*B^2*b^3*c*d^8*e - 1512*A^2*b^3*c*d^6*e^3 - 256*A*B*b*c^3*d^9 + 960*A*B*b^2*c^2*d^8*e
- 1200*A*B*b^3*c*d^7*e^2))/(64*b^10))^(1/2) - (3*(2700*B^3*b^13*d^4*e^13 - 18432*A^3*c^13*d^14*e^3 - 381312*A
^3*b^2*c^11*d^12*e^5 + 610560*A^3*b^3*c^10*d^11*e^6 - 562968*A^3*b^4*c^9*d^10*e^7 + 293112*A^3*b^5*c^8*d^9*e^8
- 84483*A^3*b^6*c^7*d^8*e^9 + 23868*A^3*b^7*c^6*d^7*e^10 - 11943*A^3*b^8*c^5*d^6*e^11 + 2331*A^3*b^9*c^4*d^5*
e^12 + 54*A^3*b^10*c^3*d^4*e^13 + 189*A^3*b^11*c^2*d^3*e^14 + 2304*B^3*b^3*c^10*d^14*e^3 - 10944*B^3*b^4*c^9*d
^13*e^4 + 17856*B^3*b^5*c^8*d^12*e^5 - 14328*B^3*b^6*c^7*d^11*e^6 + 18828*B^3*b^7*c^6*d^10*e^7 - 30672*B^3*b^8
*c^5*d^9*e^8 + 21060*B^3*b^9*c^4*d^8*e^9 - 6696*B^3*b^10*c^3*d^7*e^10 + 9252*B^3*b^11*c^2*d^6*e^11 + 4725*A*B^
2*b^13*d^3*e^14 + 129024*A^3*b*c^12*d^13*e^4 - 9360*B^3*b^12*c*d^5*e^12 - 13824*A*B^2*b^2*c^11*d^14*e^3 + 7603
2*A*B^2*b^3*c^10*d^13*e^4 - 158976*A*B^2*b^4*c^9*d^12*e^5 + 167184*A*B^2*b^5*c^8*d^11*e^6 - 143316*A*B^2*b^6*c
^7*d^10*e^7 + 184869*A*B^2*b^7*c^6*d^9*e^8 - 187434*A*B^2*b^8*c^5*d^8*e^9 + 93987*A*B^2*b^9*c^4*d^7*e^10 - 356
40*A*B^2*b^10*c^3*d^6*e^11 + 34803*A*B^2*b^11*c^2*d^5*e^12 - 172800*A^2*B*b^2*c^11*d^13*e^4 + 437184*A^2*B*b^3
*c^10*d^12*e^5 - 578448*A^2*B*b^4*c^9*d^11*e^6 + 471420*A^2*B*b^5*c^8*d^10*e^7 - 361773*A^2*B*b^6*c^7*d^9*e^8
+ 350001*A^2*B*b^7*c^6*d^8*e^9 - 253071*A^2*B*b^8*c^5*d^7*e^10 + 90423*A^2*B*b^9*c^4*d^6*e^11 - 12798*A^2*B*b^
10*c^3*d^5*e^12 + 4104*A^2*B*b^11*c^2*d^4*e^13 - 22410*A*B^2*b^12*c*d^4*e^13 + 27648*A^2*B*b*c^12*d^14*e^3 - 1
890*A^2*B*b^12*c*d^3*e^14))/(32*b^12*c^5)))*((9*(256*A^2*c^4*d^9 + 64*B^2*b^2*c^2*d^9 + 441*A^2*b^4*d^5*e^4 +
144*B^2*b^4*d^7*e^2 + 1968*A^2*b^2*c^2*d^7*e^2 + 504*A*B*b^4*d^6*e^3 - 1152*A^2*b*c^3*d^8*e - 192*B^2*b^3*c*d^
8*e - 1512*A^2*b^3*c*d^6*e^3 - 256*A*B*b*c^3*d^9 + 960*A*B*b^2*c^2*d^8*e - 1200*A*B*b^3*c*d^7*e^2))/(64*b^10))
^(1/2)*2i + (((d + e*x)^(1/2)*(7*B*b^6*d^2*e^6 - 24*A*c^6*d^7*e + 84*A*b*c^5*d^6*e^2 - 3*A*b^5*c*d^2*e^6 - 20*
B*b^5*c*d^3*e^5 - 102*A*b^2*c^4*d^5*e^3 + 45*A*b^3*c^3*d^4*e^4 - 33*B*b^2*c^4*d^6*e^2 + 24*B*b^3*c^3*d^5*e^3 +
10*B*b^4*c^2*d^4*e^4 + 12*B*b*c^5*d^7*e))/(4*b^4) - ((d + e*x)^(3/2)*(14*B*b^6*d*e^6 - 72*A*c^6*d^6*e + 216*A
*b*c^5*d^5*e^2 - 49*B*b^5*c*d^2*e^5 - 217*A*b^2*c^4*d^4*e^3 + 74*A*b^3*c^3*d^3*e^4 + 5*A*b^4*c^2*d^2*e^5 - 81*
B*b^2*c^4*d^5*e^2 + 41*B*b^3*c^3*d^4*e^3 + 39*B*b^4*c^2*d^3*e^4 - 6*A*b^5*c*d*e^6 + 36*B*b*c^5*d^6*e))/(4*b^4)
+ ((d + e*x)^(7/2)*(9*B*b^5*c*e^5 + 24*A*c^6*d^4*e - 5*A*b^4*c^2*e^5 - 48*A*b*c^5*d^3*e^2 + 3*A*b^3*c^3*d*e^4
- 19*B*b^4*c^2*d*e^4 + 21*A*b^2*c^4*d^2*e^3 + 15*B*b^2*c^4*d^3*e^2 + 3*B*b^3*c^3*d^2*e^3 - 12*B*b*c^5*d^4*e))
/(4*b^4) + ((d + e*x)^(5/2)*(7*B*b^6*e^6 - 3*A*b^5*c*e^6 - 72*A*c^6*d^5*e + 180*A*b*c^5*d^4*e^2 + 10*A*b^4*c^2
*d*e^5 - 136*A*b^2*c^4*d^3*e^3 + 24*A*b^3*c^3*d^2*e^4 - 63*B*b^2*c^4*d^4*e^2 + 14*B*b^3*c^3*d^3*e^3 + 48*B*b^4
*c^2*d^2*e^4 + 36*B*b*c^5*d^5*e - 38*B*b^5*c*d*e^5))/(4*b^4))/(c^5*(d + e*x)^4 - (d + e*x)*(4*c^5*d^3 + 2*b^2*
c^3*d*e^2 - 6*b*c^4*d^2*e) - (4*c^5*d - 2*b*c^4*e)*(d + e*x)^3 + c^5*d^4 + (d + e*x)^2*(6*c^5*d^2 + b^2*c^3*e^
2 - 6*b*c^4*d*e) + b^2*c^3*d^2*e^2 - 2*b*c^4*d^3*e) + (2*B*e^4*(d + e*x)^(1/2))/c^3
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)**(9/2)/(c*x**2+b*x)**3,x)
[Out]
Timed out
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